text:
\section{Introduction}

In this paper, we study in detail a $2$-categorical generalization of the construction of the category of elements, that we call the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction}. Being this also a natural extension of the usual Grothendieck construction (to admit $2$-copresheaves $\catfont{B}\to\catfont{CAT}$ with $\catfont{B}$ a $2$-category), the results presented here apply to the latter as well. An explicit definition of the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction} firstly appeared in Street's~\cite{street_limitsindexedbycatvalued}, but we present it in a more abstract way, that is particularly relevant to, but not only, (higher dimensional) elementary topos theory. Furthermore, after showing a connection between the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction} and what we call the \predfn{\pteor{op}lax normal $2$-\pteor{co}limits}, we use the latter to state and prove a pointwise Kan extension result for the former.

In dimension $1$, the construction of the {category of elements}, that we shall think of as the $\catfont{Set}$-enriched Grothendieck construction, is a well known process (for which we take as main reference Section 1.5 of Mac Lane and Moerdijk's~\cite{maclanemoerdijk_sheavesingeoandlogic}) of reorganization of the data of a (co)presheaf in terms of a {discrete (op)fibration}, that forms an equivalence of categories. Such construction brings many advantages, as it allows to consider a family of sets indexed over a category without mentioning morphisms that land into the universe of all sets, and it also makes the change of base operation (written in the assignment on morphisms of the (co)presheaf) more evident. But also the process itself gives important consequences, among which the reduction of the weighted $\catfont{Set}$-enriched limits to conical ones (that also brings to the explicit formula for the $\catfont{Set}$-enriched Kan extensions). Surprisingly, more is true. Indeed the wish of conicalizing the $\catfont{Set}$-enriched limits actually guides and almost determines the definition of the construction of the category of elements. And we will show how the wish of giving an essential solution to the problem of conicalizing weighted $2$-limits guides the definition of the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction} (\nameit{Construction}\ref{2SetGrothconstrtoconicalize}), justifying the explicit definition given by Street in~\cite{street_limitsindexedbycatvalued} (and the definition of the usual Grothendieck construction as well).

This will also bring to the \BF{first main result} of this paper (\nameit{Theorem}\ref{redlaxnormalconical}), which is a new, more intuitive and elementary proof of the fact that every weighted $2$-limit can be essentially conicalized, up to admitting $2$-cells inside the cones and using what we call the \predfn{lax normal conical $2$-limits} (firstly proved by Street in~\cite{street_limitsindexedbycatvalued}). The \predfn{lax normal conical $2$-limits}, that are actually a particular case of a $2$-dimensional notion of limit introduced by Gray in~\cite{gray_formalcattheory}, can be very useful, giving a theory that is equivalent to but in many situations simpler than the one of weighted $2$-limits. Indeed one can sometimes handle $2$-cells inside the cones quite easily, but really needs to have one selected universal cone rather than a bunch of cones forming a cylinder. Despite their potential, these limits have been almost forgotten, at least until Descotte, Dubuc and Szyld's paper~\cite{descottedubucszyld_sigmalimandflatpseudofun} (where they use their pseudo version, called by them \predfn{sigma-limits}).

\begin{defne}\label{defexplicgrothconstr}
Explicitly, in dimension $1$, the {construction of the category of elements} is based on the idea of taking the disjoint union of a family of sets in order to reorganize such sets as fibres of a unique function into the set of indexes. But this idea is extended to take into account also families of sets that are indexed over a category (rather than just a set), building a notion of morphism between objects of the disjoint union as a property on the morphisms between indexes. Indeed, a functor $F\:\catfont{B}\to\catfont{Set}$ is reorganized as a functor $\groth{F}\:\Grothop{F}\to \catfont{B}$, that we call the \dfn{construction of the category of elements, or the $\catfont{Set}$-enriched Grothendieck construction, of $F$}, with $\Grothop{F}$ such that:
\begin{description}
	\item[an object of $\Grothop{F}$] is a pair $(B,X)$ with $B\in \catfont{B}$ and $X\in F(B)$;
	\item[a morphism $(B,X)\to (C,X')$] is a morphism $f\:B\to C$ in $\catfont{B}$ such that $F(f)(X)=X'$.
\end{description}
Analogously, in dimension $2$, the idea of taking some disjoint union of a family of categories indexed over a category is realized by the usual Grothendieck construction (see Section 1.1 of Jacobs's book~\cite{jacobs_catlogicandtypetheory}). But we can then extend it to take into account also families of categories that are indexed over a $2$-category (rather than just a category), building a notion of $2$-cell in the corresponding $\Grothop{F}$ as a property on the $2$-cells in the category of indexes. A $2$-functor $F\:\catfont{B}\to\catfont{CAT}$ with $\catfont{B}$ a $2$-category is then reorganized as a $2$-functor $\groth{F}\:\Grothop{F}\to \catfont{B}$, that we call the \dfn{$2$-$\catfont{Set}$-enriched Grothendieck construction of $F$}, with $\Grothop{F}$ such that:
\begin{description}
	\item[an object of $\Grothop{F}$] is a pair $(B,X)$ with $B\in \catfont{B}$ and $X\in F(B)$;
	\item[a morphism $(B,X)\to (C,X')$ in $\Grothop{F}$] is a pair $(f,\alpha)$ with $f\:B\to C$ a morphism in $\catfont{B}$ and $\alpha\:F(f)(X)\to X'$ a morphism in $F(B)$;
	\item[a $2$-cell $(f,\alpha)\aR{}(g,\beta)\:(B,X)\to (C,X')$] is a $2$-cell $\delta\:f\aR{}g$ in $\catfont{B}$ such that $\alpha=\beta\circ F(\delta)_X$.
\end{description}
\end{defne}

In dimension $1$, it is then known that the construction of the category of elements can be captured in a more abstract way, which is particularly interesting for the ($2$-dimensional) elementary topos theory.

\begin{prop}\label{catofelementsiscomma}
	Let $F\:\catfont{B}\to \catfont{Set}$ be a copresheaf. The construction of the category of elements of $F$ is equivalently given by the comma object
 	\begin{eqD}{catofelemcomma}
 		\sq*[l][7][7][\opn{comma}][2.7][2.2][0.6]{\Grothop{F}}{\catfont{1}}{\catfont{B}}{\catfont{Set}}{}{\groth{F}}{1}{F}
 	\end{eqD}
 	As a direct consequence, it is also given by the pullback between $F$ and the lax limit of the arrow $1\:\catfont{1}\to \catfont{Set}$, where the latter coincides with the forgetful functor $\tau\:\catfont{Set}_{\bullet}\to \catfont{Set}$ from pointed sets to sets.
 	\begin{eqD*}
 		\begin{cd}*
 			\Grothop{F}\PB{rd} \arrow[d,"\groth{F}"'] \arrow[r] \& \catfont{Set}_{\bullet} \arrow[d,"{\tau}"'] \arrow[r]\& \catfont{1} \arrow[d,"{1}"]\arrow[ld,Rightarrow,shorten <=2.7ex,shorten >=2.2ex,"\opn{comma}"{pos=0.6}] \\
 			\catfont{B}\arrow[r,"F"'] \& \catfont{Set} \arrow[r,equal]\& \catfont{Set}
 		\end{cd}
 	\end{eqD*}
\end{prop}

According to Weber's~\cite{weber_yonfromtwotop}, \nameit{Proposition}\ref{catofelementsiscomma} shows the construction of the category of elements as the archetypal $2$-dimensional classification process, exhibiting $\catfont{Set}$ as the canonical ($2$-dimensional) universe for the $2$-category $\catfont{CAT}$. What gets classified is precisely the discrete opfibrations with small fibres. Weber actually only considered the point of view of the second part of \nameit{Proposition}\ref{catofelementsiscomma}, but we prefer the first, that of equation~\refs{catofelemcomma}, as it shows the singleton as the verum inside the category $\catfont{Set}$ of generalized truth values. Equation~\refs{catofelemcomma} also makes it clear that the passage from the $1$-dimensional elementary topos theory to the $2$-dimensional one is obtained by upgrading the classification by pullbacks to one by comma objects. As a consequence, we end up classifying discrete opfibrations, since they are (defined by representability from) what is classified in the archetypal elementary $2$-topos $\catfont{CAT}$.

We are interested in understanding a general pattern, for example from an enriched point of view. Of course it is natural to consider $1\:\catfont{1}\to \catfont{Set}$ as the verum truth value, and this has an immediate generalization to the general enriched setting, but we would like to understand the reason why we should take the comma object to regulate the classification process in dimension $2$. What we believe is the deep reason behind this is that the comma objects are the archetypal example of \predfn{exact square in \catfont{CAT}} (\nameit{Definition}\ref{defexactsquare}), with the notable consequence that comma objects manage to express every copresheaf as a pointwise left Kan extension of the constant copresheaf at $1$ (\nameit{Theorem}\ref{copresheavesarekanextensions}).

\begin{defne}\label{defexactsquare}
	An \dfn{exact square in \catfont{CAT}} is a diagram
	\sq[l][7][7][\lambda][2.7][2.2]{\catfont{L}}{\catfont{A}}{\catfont{B}}{\catfont{C}}{g}{p}{q}{f}
	in \catfont{CAT} such that, for every category $\catfont{M}$, the associated {Beck-Chevalley natural transformation} given by the pasting
	\begin{cd}
		\& \m{\catfont{L}}{\catfont{M}} \arrow[d,Rightarrow,shift right = 6.5ex,shorten <= 1ex, shorten >= 1ex,"{\epsilon^p}"'] \arrow[ld,bend right,"{\Lan{p}}"'] \arrow[r,leftarrow,"{-\circ g}"] \arrow[d,leftarrow,"{-\circ p}"'] \& \m{\catfont{A}}{\catfont{M}} \arrow[d,Rightarrow,shift left = 7ex,shorten <= 1ex, shorten >= 1ex,"{\eta^q}"] \arrow[d,leftarrow,"{-\circ q}"] \arrow[ld,Rightarrow,shorten <=2.7ex,shorten >=2.2ex,"{-\ast \lambda}"{pos=0.5}] \arrow[r,equal,"{}"] \& \m{\catfont{A}}{\catfont{M}} \arrow[ld,bend left,"{\Lan{q}}"]\\
		\m{\catfont{B}}{\catfont{M}} \arrow[r,equal,"{}"] \& \m{\catfont{B}}{\catfont{M}} \arrow[r,leftarrow,"{-\circ f}"'] \& \m{\catfont{C}}{\catfont{M}}
	\end{cd}
	is an isomorphism, where $\Lan{p}$ and $\Lan{q}$ denote the pointwise left Kan extensions respectively along $p$ and along $q$, and $\epsilon^p$ and $\eta^q$ are respectively the counit of the adjunction formed by the first one and the unit of the adjunction formed by the second one.
\end{defne}

\begin{teor}\label{copresheavesarekanextensions}
	Let $F\:\catfont{B} \to \catfont{Set}$ be a copresheaf. Then
	$$F=\Lan{\groth{F}}{\Delta 1}$$
	where $\Delta 1\:\Grothop{F}\to \catfont{Set}$ is the functor constant at $1$.
\end{teor}
\begin{proof}
	This proof does not seem to appear in the literature, but might be folklore. By \nameit{Proposition}\ref{catofelementsiscomma}, the construction of the category of elements can be expressed as a comma object, forming thus an exact square. Considering $\catfont{M}=\catfont{Set}$ in the definition of exact square, the component of the associated Beck-Chevalley transformation on the copresheaf $1\:\catfont{1}\to \catfont{Set}$ exhibits
	$$\Lan{\groth{F}}{\Delta 1}\cong \Lan{1}{1} \circ F.$$
	And since $1\:\catfont{1}\to \catfont{Set}$ is dense, we have that $\Lan{1}{1}=\Id{\catfont{Set}}$ (a reference for this basic result on density is Kelly's~\cite{kelly_basicconceptsofenriched}).
\end{proof}

\begin{rem}\label{remtheorkanextisuseful}
	\nameit{Theorem}\ref{copresheavesarekanextensions} (together with \nameit{Proposition}\ref{catofelementsiscomma}) implies a huge portion of the theory around the construction of the category of elements, including the canonical extension of the construction to a functor $\groth{-}\:\m{\catfont{B}}{\catfont{Set}}\to \slice{\catfont{Cat}}{\catfont{B}}$, the fully faithfulness of $\groth{-}$, the stability under pullback of the discrete (op)fibrations with small fibres (that form the essential image of $\groth{-}$) and the conicalization of the weighted $\catfont{Set}$-enriched limits. This does not seem to appear in the literature, but might be folklore.

	For example, the extension of the construction of the category of elements to a fully faithful functor $\groth{-}\:\m{\catfont{B}}{\catfont{Set}}\to \slice{\catfont{Cat}}{\catfont{B}}$ can be obtained by the chain of isomorphisms
	$$\HomC{\m{\catfont{B}}{\catfont{Set}}}{F}{G}\cong \HomC{\m{\Grothopdiag{F}}{\catfont{Set}}}{\Delta 1}{G\circ \groth{F}}\cong \HomC{\slice{\catfont{Cat}}{\catfont{B}}}{\Grothopdiag{F}}{\Grothopdiag{G}}$$
	(for $F$ and $G$ arbitrary), where the first isomorphism is given by $F=\Lan{\groth{F}}{\Delta 1}$ and the second one is given by the universal property of the comma object $\Grothop{G}$.
\end{rem}

The \BF{second} (\nameit{Theorem}\ref{grothconstrislaxcomma}) and \BF{third} (\nameit{Theorem}\ref{teorpointkanextforgroth}) \BF{main results} of this paper can be condensed in the following theorem, that gives a categorification of both \nameit{Proposition}\ref{catofelementsiscomma} and \nameit{Theorem}\ref{copresheavesarekanextensions}. Exactly as its $1$-dimensional analogue yields a lot of useful applications (see \nameit{Remark}\ref{remtheorkanextisuseful}), so can this result.

\begin{teor}\label{theorcondensed}
	Let $F\:\catfont{B}\to \catfont{CAT}$ be a $2$-functor. The \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction} is equivalently given by the \predfn{lax comma object} \pteor{with a new universal property, described in \nameit{Definition}\ref{completeunivproplaxcomma}, that refines both the ones given by Gray in~\cite{gray_formalcattheory} and by Lambert in~\cite{lambert_discretetwofib}}
	\begin{eqD*}
		\sq*[l][7][7][\lax \opn{comma}][2.7][2.2][0.65]{\Grothop{F}}{\catfont{1}}{\catfont{B}}{\catfont{CAT}}{}{\groth{F}}{\catfont{1}}{F}
	\end{eqD*}
	Furthermore, this square \pteor{filled with a \predfn{lax normal natural transformation}} exhibits $F$ as the \predfn{pointwise left Kan extension of $\Delta 1$ along $\groth{F}$ in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$} \pteor{it lives in a tridimensional world that admits the lax natural transformations, see \nameit{Remark}\ref{needof2catlax}}, where this concept of pointwise Kan extension is defined originally in this paper in \nameit{Definition}\ref{defpointkanextensionlaxthreecat}, using the concept of \pteor{non necessarily conical anymore} \predfn{oplax normal $2$-colimit} that we introduce in Section~\ref{sectionlaxnormallimits}.
	$$F=\Lan{\groth{F}}{\Delta 1}.$$
\end{teor}

\nameit{Theorem}\ref{theorcondensed} shows in which sense the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction} can be thought of as the archetypal $3$-dimensional classification process, in the sense of a would-be tridimensional elementary topos theory. What gets classified are the \predfn{split $2$-$\catfont{Set}$-opfibrations} with small fibres (introduced in Lambert's~\cite{lambert_discretetwofib}) as \predfn{discrete $2$-fibrations}). Analogously to how the classification by pullbacks (in dimension $1$) is upgraded to one by comma objects to obtain the notion of $2$-dimensional elementary topos (see above), we believe that it should then be upgraded to one by lax comma objects (defined as in \nameit{Definition}\ref{completeunivproplaxcomma}), in order to reach a definition of \predfn{tridimensional elementary topos}. It is interesting to notice that we have to move out of $2\h[2]\mbox{-}\CAT$ (see \nameit{Remark}\ref{needof2catlax}) in order to capture the laxness that permeates the Grothendieck construction. So the archetypal elementary $3$-topos seems to be $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$, inscribed in a sequence
$$\catfont{Set}\quad\leadsto\quad \catfont{CAT} \quad\leadsto\quad 2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$$
that we believe is best explained by what we call a \predfn{$2$-$\catfont{V}$-enrichment} (\nameit{Definition}\ref{deftwovenrichment}):
$$\catfont{V}\quad\leadsto\quad \VCAT{\catfont{V}} \quad\leadsto\quad \twoVCAT{\catfont{V}}$$
(whence comes the name we give to the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction}).

The \BF{fourth main result} of this paper (\nameit{Proposition}\ref{pointkanisalsoweak}) is that a \predfn{pointwise left Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$} (\nameit{Definition}\ref{defpointkanextensionlaxthreecat}) along a \predfn{$2$-$\catfont{Set}$-opfibration} (\nameit{Definition}\ref{deftwosetopf}) is always also a \predfn{weak left Kan extension} (\nameit{Definition}\ref{defweakkanextensionlaxthreecat}). This justifies and explains more the concept of pointwise Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ that we propose here. Such result is based on a new $\oplaxn\mbox{-}\lax$ generalization of the parametrized Yoneda lemma, presented in \nameit{Theorem}\ref{parametrizedyonedaoplaxnlax}, that shows how the oplax normal naturality is the minimum amount of strictness required to expand the data on the identities to a lax natural transformation.

\subsection*{Outline of the paper}

In Section~\ref{sectionlaxnormallimits}, we produce, simultaneously (\nameit{Construction}\ref{2SetGrothconstrtoconicalize}), both the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction} and the \predfn{lax normal conical $2$-limits}. We show with new proofs that the lax normal conical $2$-limits and the weighted $2$-limits give equivalent theories (\nameit{Theorem}\ref{redlaxnormalconical}, \nameit{Theorem}\ref{laxnormalareweighted}) and consider a colimit version as well. In Section~\ref{sectiontwosetgrothconstr}, we refine the universal property of the lax comma object to suit a lax $3$-categorical ambient (\nameit{Definition}\ref{completeunivproplaxcomma}) and we conceive the $2$-$\catfont{Set}$-enriched Grothendieck construction as the archetypal $3$-dimensional classifier (\nameit{Theorem}\ref{grothconstrislaxcomma}). We then inscribe this in a weak enrichment idea (\nameit{Remark}\ref{remarchetypalthreedimclassifproc}). In Section~\ref{sectionkanextensions}, we present and apply a pointwise Kan extension result for the $2$-$\catfont{Set}$-enriched Grothendieck construction (\nameit{Theorem}\ref{teorpointkanextforgroth}), after giving an original definition of \predfn{pointwise Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$} (\nameit{Definition}\ref{defpointkanextensionlaxthreecat}). We conclude by showing that a pointwise Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ is always a weak one as well (\nameit{Proposition}\ref{pointkanisalsoweak}), using a $\oplaxn\mbox{-}\lax$ generalization of the parametrized Yoneda lemma (\nameit{Theorem}\ref{parametrizedyonedaoplaxnlax}).

\section{(Op)lax normal conical 2-(co)limits}
\label{sectionlaxnormallimits}

In this section, we show how the wish of giving an essential solution to the problem of conicalization of the weighted $2$-limits produces, simultaneously, both the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction}, thus justifying the explicit definition of \nameit{Definition}\ref{defexplicgrothconstr} that Street gave in~\cite{street_limitsindexedbycatvalued}, and the \predfn{lax normal conical $2$-limits}, that are a particular case of a $2$-dimensional limit introduced by Gray in~\cite{gray_formalcattheory}. This brings to the first main result of this paper (\nameit{Theorem}\ref{redlaxnormalconical}), which is a new, more intuitive and elementary proof of the fact (firstly proved by Street in~\cite{street_limitsindexedbycatvalued}) that every weighted $2$-limit can be reduced to a \predfn{lax normal conical} one - that is, one of conical form but with coherent $2$-cells inside - using the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction}.

These two kinds of $2$-limit give equivalent theories, as it is also true that a \predfn{lax normal conical $2$-limit} is a weighted one - proved here with a new more explicit proof or originally by Street in~\cite{street_limitsindexedbycatvalued} - but \predfn{lax normal conical} ones are in many situations simpler to use. Indeed one can sometimes handle $2$-cells inside the cones quite easily, but really needs to have one selected universal cone rather than a bunch of cones forming a cylinder.

The following idea may be kept in mind to understand the differences between these two kinds of $2$-limit. When one moves to dimension $2$, the conical limits do not suffice anymore, as the functors from the terminal $\catfont{1}$ to a category $\catfont{C}$ cannot capture the whole of $\catfont{C}$, but just its objects. A solution to this problem is to consider at least also the functors from $\catfont{2}$ to $\catfont{C}$ (or all the functors from anything to $\catfont{C}$), in order to capture the morphisms of $\catfont{C}$ as well; this brings to the concept of weighted $2$-limit. But there is another solution, that is considering the functors from the terminal $\catfont{1}$ to $\catfont{C}$ and now also the natural transformations between them; this brings towards the concept of \predfn{lax normal conical $2$-limit}.

We then notice that, in order to essentially conicalize the weighted $2$-colimits, it is more natural to use the \predfn{oplax normal conical $2$-colimits}. It is the non necessarily conical anymore \predfn{oplax normal $2$-colimits} that we will use in Section~\ref{sectionkanextensions} to propose a notion of colimit in a \predfn{$2$-$\catfont{Set}$-category} and with that one of \predfn{pointwise left Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$}.

\begin{rec}
	Let $\catfont{V}$ to be a complete and cocomplete symmetric closed monoidal category. Let $F\:\catfont{A} \to \catfont{C}$ be a $\catfont{V}$-functor with $\catfont{A}$ a small $\catfont{V}$-category, and let $W\:\catfont{A}\to \catfont{V}$ be another $\catfont{V}$-functor (called the \dfn{weight}, in place of the classical constant at $1$ functor $\Delta 1$, that now does no longer suffice). The \dfn{$\catfont{V}$-limit of $F$ weighted by $W$}, denoted as $\wlim{W}{F}$, is (if it exists) an object $L\in \catfont{C}$ together with an isomorphism in \catfont{V}
	\begin{equation}\label{isoweightlim}
		\HomC{\catfont{C}}{U}{L}\cong \HomC{\m{\catfont{A}}{\catfont{V}}}{W}{\HomC{\catfont{C}}{U}{F(-)}}
	\end{equation}
	$\catfont{V}$-natural in $U\in \catfont{C}\ensuremath{^{\operatorname{op}}}$, where $\m{\catfont{A}}{\catfont{V}}$ is the $V$-category of $\catfont{V}$-copresheaves on $\catfont{A}$ valued in $\catfont{V}$ enriched over itself. (For example, if $\catfont{V}=\catfont{Cat}$, then $\m{\catfont{A}}{\catfont{V}}$ is the $2$-category of $2$-functors, $2$-natural transformations and modifications from $\catfont{A}$ to \catfont{CAT}.) Notice that, when $\wlim{W}{F}$ exists, the isomorphism of equation~\refs{isoweightlim} with $U=L$ gives us in particular a $\catfont{V}$-natural transformation
	$$\lambda\:W\aR{}\HomC{\catfont{C}}{L}{F(-)}$$
	by considering the identity on $L$; this $\lambda$ is called the \dfn{universal cylinder}.

	Given instead $F\:\catfont{A}\to\catfont{C}$ and $W\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\catfont{V}$ two $\catfont{V}$-functors with $\catfont{A}$ small, the \dfn{$\catfont{V}$-colimit of $F$ weighted by $W$}, denoted as $\wcolim{W}{F}$, is defined by the universal property
	\begin{equation*}
		\HomC{\catfont{C}}{\wcolim{W}{F}}{U}\cong \HomC{\m{\catfont{A}\ensuremath{^{\operatorname{op}}}}{\catfont{V}}}{W}{\HomC{\catfont{C}}{F(-)}{U}}
	\end{equation*}
	$\catfont{V}$-natural in $U\in \catfont{C}$. And in place of $\lambda$ we find the \dfn{universal cocylinder}
	$$\mu\:W\aR{}\HomC{\catfont{C}}{F(-)}{C}.$$
\end{rec}

\begin{rec}\label{recstrategyconicalize}
	Although the classical constant at $1$ weight $\Delta 1$, called the \dfn{conical weight}, does no longer suffice in the general enriched setting, we pay attention to when a weighted limit can be \dfn{reduced to a conical one} (also said \dfn{conicalized}), i.e.\ reduced to one weighted by $\Delta 1$.

	It is well known (see Kelly's~\cite{kelly_basicconceptsofenriched}) that in the $\catfont{Set}$-enriched setting every weighted limit can be conicalized, using the construction of the category of elements (that we shall think of as the $\catfont{Set}$-enriched Grothendieck construction). A known strategy to show this (used by Kelly in~\cite{kelly_basicconceptsofenriched}) is to show that the particular weighted $\catfont{V}$-colimits
	$$W\cong \wcolim{W}{\yy},$$
	for $W\:\catfont{A}\to \catfont{V}$ an enriched presheaf with $\catfont{A}$ small and $\yy\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\m{\catfont{A}}{\catfont{V}}$ the Yoneda embedding (that are immediatly seen to hold by the Yoneda lemma), are conicalizable in a nice way for $\catfont{V}=\catfont{Set}$, and deduce that then every weighted $\catfont{Set}$-limit is conicalizable using the lemma of continuity of a limit in its weight, that we recall here below. The idea is that if we manage to write any weighted limit in the left hand side of equation~\refs{isolemmacont} with $W=\Delta 1$, we obtain in the right hand side its conicalization, provided that $H$ is nice enough (that is, similar enough to the Yoneda embedding) to make the limit $\wlim{H(-)}{F}$ simplify.
\end{rec}

\begin{lemm}[Continuity of a limit in its weight]\label{continuityinweight}
	Let $W\:\catfont{B}\ensuremath{^{\operatorname{op}}}\to\catfont{V}$, $H\:\catfont{B}\to\m{\catfont{A}}{\catfont{V}}$ and $F\:\catfont{A}\to\catfont{C}$ be $\catfont{V}$-functors, and suppose that $\wcolim{W}{H}$ and each $\wlim{H(B)}{F}$ with $B\in \catfont{B}$ exist. Then
	\begin{equation}\label{isolemmacont}
		\wlim{\wcolim{W}{H}}{F} \cong \wlim{W}{\left(\wlim{H(-)}{F}\right)}
	\end{equation}
	either side existing if the other does.

	The colimit version, with $W$ and $F$ as above but now with $H\:\catfont{B}\to\m{\catfont{A}\ensuremath{^{\operatorname{op}}}}{\catfont{V}}$, is
	$$\wcolim{\wcolim{W}{H}}{F} \cong \wcolim{W}{\left(\wcolim{H(-)}{F}\right)}.$$
\end{lemm}
\begin{proof}
	See Kelly's~\cite{kelly_basicconceptsofenriched}.
\end{proof}

\begin{rem}\label{needoflaxcones}
	We are interested in the problem of conicalization of the weighted $2$-limits. This is, strictly speaking, not possible, but we search for an essential solution to it nevertheless, motivated by the advantages that having a universal essential cone selected from the universal cylinder could bring. We will exploit the strategy described in \nameit{Recall}\ref{recstrategyconicalize} to give a new, more intuitive and elementary proof of the fact that every weighted $2$-limit can be reduced to a \predfn{lax normal conical} one.

	So we focus on conicalizing first the $2$-colimits
	$$W\cong \wcolim{W}{\yy},$$
	for $W\:\catfont{A}\to \catfont{CAT}$ an enriched presheaf with $\catfont{A}$ small and $\yy\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\m{\catfont{A}}{\catfont{CAT}}$ the $2$-Yoneda embedding. In particular, we want to encode the universal cocylinder
	$$\mu\:W\aR{}\HomC{\m{\catfont{A}}{\catfont{V}}}{\y{-}}{W}$$
	(given by the Yoneda lemma) in terms of a universal cocone
	$$\t{\mu}\:\Delta 1\aR{}\HomC{\m{\catfont{A}}{\catfont{V}}}{H(-)}{W}$$
	with $H$ some $\catfont{V}$-functor $\catfont{B}\to \m{\catfont{A}}{\catfont{V}}$. The difficulty we encounter is that, in dimension $2$, the components of $\mu$ are functors rather than mere functions, and a cocone is too limited to encode the extra data given by the assignments on morphisms. The idea is to then admit $2$-cells inside the cocone $\t{\mu}$ in order to encode the images under the components $\mu_A$ of $\mu$ (with $A\in \catfont{A}$) of the morphisms in $W(A)$. After all, such images are morphisms in $\HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{A}}{W}$, and thus $2$-cells between $\y{A}$ and $W$.

	This is explored in the following construction, that originally shows how both the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction} and the \predfn{lax normal conical $2$-limits} arise simultaneously from the wish to essentially solve the problem of conicalizing the weighted $2$-limits.
\end{rem}

	\begin{cons}[The $2$-$\catfont{Set}$-enriched Grothendieck constr.]\label{2SetGrothconstrtoconicalize}
		Following \nameit{Remark}\ref{needoflaxcones}, we search for a more relaxed notion of $2$-natural transformation and for a $2$-functor $H\:\catfont{B}\to \m{\catfont{A}}{\catfont{CAT}}$ such that any cocylinder
		$$\varphi\:W\aR{}\HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{-}}{U}$$
		with $U\:\catfont{A}\to \catfont{CAT}$ a $2$-functor can be encoded in terms of a relaxed $2$-natural transformation
		$$\t{\varphi}\:\Delta 1 \aR[\opn{relaxed}]{} \HomC{\m{\catfont{A}}{\catfont{CAT}}}{H(-)}{U}\:\catfont{B}\ensuremath{^{\operatorname{op}}}\to\catfont{CAT},$$
		that is a relaxed version of a cocone.

		As we will want to apply \nameit{Lemma}\ref{continuityinweight} (continuity of a limit in its weight) to deduce that every weighted $2$-limit can then be analogously conicalized, we search for $H$ of the form
		$${\left(\Grothopdiag{W}\right)}\ensuremath{^{\operatorname{op}}}\arr{{\groth{W}}\ensuremath{^{\operatorname{op}}}} \catfont{A}\ensuremath{^{\operatorname{op}}}\arr{\yy} \m{\catfont{A}}{\catfont{CAT}},$$
		where $\Grothop{W}$ and $\groth{A}$ are, up to now, just symbols, but will be found to be the \predfn{$2$-$\catfont{Set}$-enriched Grothendieck construction} (as defined explictly by Street in~\cite{street_limitsindexedbycatvalued}, see \nameit{Definition}\ref{defexplicgrothconstr}). Indeed, we will need the limit $\wlim{H(-)}{F}$ of the right hand side of equation~\refs{isolemmacont} to simplify, and we can achieve this using the Yoneda lemma if $H$ factorizes through the Yoneda embedding.

		For every $A\in \catfont{A}$ and $X\in W(A)$, we have a morphism
		$$\varphi_A(X)\:\y{A}\to U,$$
		and we want to form the cocone $\t{\varphi}$ exactly with these morphisms. So, for every $A\in \catfont{A}$ and $X\in W(A)$ we need an object in $\Grothop{W}$ whose image with respect to $\groth{W}$ is $A$. We call such object $(A,X)$. Since these are the only objects we need for our conicalization process, we take the objects of $\Grothop{W}$ to be precisely all the pairs $(A,X)$ with $A\in \catfont{A}$ and $X\in W(A)$ and define $\groth{W}$ on objects to be the projection on the first component. We then take
		$$\t{\varphi}_{(A,X)}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \varphi_A(X).$$

		But we also need to encode inside $\t{\varphi}$ the assignment of every $\varphi_A$ with $A\in \catfont{A}$ on morphisms $\alpha\:X\to X'$ in $W(A)$. And, after \nameit{Remark}\ref{needoflaxcones}, the idea is to use a relaxed version of a cocone. We would like $\t{\varphi}$ to be at least a lax natural transformation, so that it is not ill-behaved. So, for every morphism $\xi\:(A,X) \to (A',X')$ in $\Grothop{W}$, we can have a $2$-cell
		\begin{cd}[6][7]
			1 \arrow[r,"{\t{\varphi}_{(A,X)}}"] \arrow[d,equal]\& \HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{A}}{U}\arrow[d,"{-\circ \y{\h\groth{W}\ensuremath{^{\operatorname{op}}}(\xi)}}"]\arrow[dl,Rightarrow,"{\t{\varphi}_\xi}"{pos=0.445}, shorten <= 2.8ex, shorten >= 4.9ex]\\
			1 \arrow[r,"{\t{\varphi}_{(A',X')}}"'] \& \HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{A'}}{U}
		\end{cd}
		For every $A\in \catfont{A}$ and every morphism $\alpha\:X\to X'$ in $W(A)$, we need a morphism $(A,X)\to (A,X')$ in $\Grothop{W}$ whose image with respect to $\groth{W}$ is $\id{A}$, so that we can take the component of $\t{\varphi}$ on it to be $\varphi_A(\alpha)$. Wishing to write the action of $\groth{W}$ again as a projection on the first component, we call such morphism $(A,X)\to (A,X')$ as $(\id{A},\alpha)$.

		Now, we want to encode the $2$-naturality of $\varphi$ into the relaxed naturality of $\t{\varphi}$. For every morphism $f\:A\to A'$ in $\catfont{A}$ and every $X\in W(A)$, the naturality of $\varphi$ expresses the equality
		$$\varphi_{A'}(W(f)(X))=\varphi_A(X)\circ \y{f}.$$
		So, for every $f\:A\to A'$ in $\catfont{A}$ and $X\in W(A)$, we need a morphism
		$$\underline{f}^X\:(A,X)\to (A',W(f)(X))$$
		in $\Grothop{W}$ such that $\groth{W}\left(\underline{f}^X\right)=f$ and $\t{\varphi}_{\underline{f}^X}=\id{}$.

		It is now natural to take $\underline{\id{A}}^X=(\id{A},\id{X})$ for every $A\in \catfont{A}$ and $X\in W(A)$ and ask any of such equal morphisms to be the identity on $(A,X)$. We will see below that the two kinds of morphisms $(\id{A},\alpha)$ and $\underline{f}^X$ are enough for our needs, but we need to close the union of these morphisms under composition. Of course, it is clear how to compose morphisms of the same kind with each other in a natural way. For the composition of different kinds of morphisms, we notice that, given $f\:A\to A'$ in $\catfont{A}$ and $\alpha\:X\to X'$ in $W(A)$, the two morphisms $\underline{f}^{X'}\circ (\id{A},\alpha)$ and $(\id{A'},W(f)(\alpha))\circ \underline{f}^X$ in $\Grothop{W}$ will have the same associated structure $2$-cell of $\t{\varphi}$, by lax naturality of $\t{\varphi}$, since by naturality of $\varphi_A$
		$$\varphi_A(\alpha)\circ \y{f}=\varphi_{A'}(W(f)(\alpha)).$$
		We then take such two morphisms in $\Grothop{W}$ to be equal, so that we will be able to recover the naturality of $\varphi$ (on morphisms) starting from $\t{\varphi}$. At this point, every finite composition of morphisms in $\Grothop{W}$ can be reduced to a composite
		$$(A,X)\ar{\underline{f}^X} (A',W(f)(X)) \ar{(\id{A'},\alpha)} (A',X')$$
		for some $f\:A\to A'$ in $\catfont{A}$ and $\alpha\:W(f)(X)\to X'$ in $W(A')$. So we define the morphisms in $\Grothop{W}$ to be precisely all the formal composites $(\id{A'},\alpha)\circ \underline{f}^X$, that we call $(f,\alpha)$, with $f\:A\to A'$ in $\catfont{A}$ and $\alpha\:W(f)(X)\to X'$ in $W(A')$. And we take the identities and the composition as described above. In particular, we see that $\underline{f}^X=(f,\id{W(f)(X)})$, whence we shall use the right hand side notation for such morphism from now on. And we can also get an explicit\v formula for the composition of an arbitrary diagram $(A,X)\ar{(f,\alpha)}(A',X')\ar{(f',\alpha')}(A'',X'')$ in $\Grothop{W}$, that is\v[-1]
		\begin{center}
			\linesep{1.5}
			\begin{tabular}{LL}
				(f',\alpha')\circ (f,\alpha)=(\id{A''},\alpha')\circ (f',\id{})\circ (\id{A'},\alpha)\circ (f,\id{})=\\
				=(\id{A''},\alpha')\circ (\id{A''},W(f')(\alpha))\circ (f',\id{})\circ (f,\id{})=(f'\circ f,\h \alpha'\circ W(f')(\alpha)).
			\end{tabular}
		\end{center}
		It is readily seen that we have just given the data for a category $\Grothop{W}$. Since we want $\groth{W}$ to be a functor, for an arbitrary morphism $(f,\alpha)$ in $\Grothop{W}$ we need to have
		$$\groth{W}(f,\alpha)=\groth{W}(\id{A'},\alpha)\circ \groth{W}(f,\id{})=f$$
		So, with the notation we have chosen, $\groth{W}$ is still acting as a projection on the first component. As we want $\t{\varphi}$ to be at least a lax natural transformation, the structure $2$-cell of $\t{\varphi}$ associated to an arbitrary morphism $(f,\alpha)$ in $\Grothop{W}$ needs to be
		$$\t{\varphi}_{(f,\alpha)}=\t{\varphi}_{(\id{A'},\alpha)\circ (f,\id{})}=\varphi_{A'}(\alpha)\circ \id{}=\varphi_{A'}(\alpha).$$

		We now want to encode the $2$-dimensional part of the $2$-naturality of $\varphi$ into the $2$-dimensional part of the relaxed naturality of $\t{\varphi}$. As we want $\t{\varphi}$ to be at least lax natural, we will have, for every $2$-cell $\Xi\:(f,\alpha)\aR{}(g,\beta)\:(A,X)\to (A',X')$ in $\Grothop{W}$,
		\begin{eqD}{twodimlaxphitilde}
			\begin{cd}*
				1 \arrow[r,"{\t{\varphi}_{(A,X)}}"] \arrow[d,equal]\& \HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{A}}{U}\arrow[d,"{-\circ \y{f}}"]\arrow[dl,Rightarrow,"{\t{\varphi}_{(f,\alpha)}}"{pos=0.445}, shorten <= 2.8ex, shorten >= 4.9ex]\\
				1 \arrow[r,"{\t{\varphi}_{(A',X')}}"'] \& \HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{A'}}{U}
			\end{cd} = \quad
			\begin{cd}*
				1 \arrow[r,"{\t{\varphi}_{(A,X)}}"] \arrow[d,equal]\& 	\HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{A}}{U}\arrow[d,bend right,shift right=0ex,"{-\circ \y{g}}"'{name=B}]\arrow[d,bend left,shift left = 2ex,"{-\circ \y{f}}"{name=A}]\arrow[dl,Rightarrow,shift right=1ex,"{\t{\varphi}_{(g,\beta)}}"{pos=0.55}, shorten <= 4.65ex, shorten >= 3.2ex]\\
				1 \arrow[r,"{\t{\varphi}_{(A',X')}}"'] \& \HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{A'}}{U}
				\arrow[Rightarrow,from=A,to=B,"{-\ast \y{\h\groth{W}\ensuremath{^{\operatorname{op}}}(\Xi)}}"'{inner sep=1ex},shorten <=1.55ex, shorten >=1.5ex]
			\end{cd}
		\end{eqD}
		The $2$-naturality of $\varphi$ expresses the following equality, for every $2$-cell $\delta\:f\aR{}g\:A\to A'$ in $\catfont{A}$ and for every $X\in W(A)$:
		$$\varphi_{A'}(W(\delta)_X)=\varphi_A(X)\h\y{\delta}\:\varphi_{A'}(W(f)(X))\aR{}\varphi_{A'}(W(g)(X)).$$
		So, for every $2$-cell $\delta\:f\aR{}g\:A\to A'$ in $\catfont{A}$ and every $X\in W(A)$, we need a $2$-cell in $\Grothop{W}$, that we call $\underline{\delta}^X$ or just $\delta$, such that
		$$\underline{\delta}^X\:(f,W(\delta)_X)\aR{}(g,\id{})\:(A,X)\to (A',W(g)(X))$$
		and $\groth{W}(\underline{\delta}^X)=\delta$. These are the only $2$-cells in $\Grothop{W}$ that we need in order to encode the $2$-naturality of $\varphi$ and they are closed under both the vertical composition and the horizontal composition inherited from $\catfont{A}$, but we have to close them under whiskering with each of the two kinds of $1$-cells in $\Grothop{W}$. The horizontal composition of those $2$-cells inherited from $\catfont{A}$ tells us in particular how to whisker with morphisms of type $(h,\id{})$. Precisely, given a $2$-cell $\delta\:f\aR{}g\:A\to A'$ and morphisms $h\:B\to A$ and $k\:A'\to C$ in $\catfont{A}$, and given $X\in W(A)$ and $Y\in W(B)$,
		$$(k,\id{})\h\underline{\delta}^X= \underline{(k\h\delta)}^X \quad\text{and}\quad\underline{\delta}^{W(h)(Y)}\h (h,\id{})= \underline{(\delta \h h)}^Y.$$
		Now, since for every $\delta\:f\aR{}g\:A\to A' $ in $\catfont{A}$ and every $\alpha\:X\to X'$ in $W(A)$, we have that the axiom of equation~\refs{twodimlaxphitilde} of $\t{\varphi}$ on the two whiskerings $\underline{\delta}^{X'}\h(\id{A},\alpha)$ and $(\id{A'},W(g)(\alpha))\h \underline{\delta}^{X}$ in $\Grothop{W}$ is exactly the same, by the analogous swapping property we asked the composition of $1$-cells in $\Grothop{W}$, we ask such two whiskerings in $\Grothop{W}$ to be equal. So, at this point, every whiskering and then every horizontal composition of $2$-cells in $\Grothop{W}$ can be reduced to a whiskering of the form $(\id{},\beta)\h \underline{\delta}^X$ for some $2$-cell $\delta\:f\aR{}g\:A\to A'$ in $\catfont{A}$, $X\in W(A)$ and $\beta\:W(g)(X)\to X'$ in $W(A')$.

		So we define the $2$-cells in $\Grothop{W}$ to be precisely the formal whiskerings $(\id{},\beta)\h \underline{\delta}^X$ for some $2$-cell $\delta\:f\aR{}g\:A\to A'$ in $\catfont{A}$, $X\in W(A)$ and $\beta\:W(g)(X)\to X'$ in $W(A')$. Equivalently, a $2$-cell $(f,\alpha)\aR{}(g,\beta)\:(A,X)\to (A',X')$ in $\Grothop{W}$ is a $2$-cell $\delta\:f\aR{}g$ in $\catfont{A}$ such that
		$$\alpha=\beta\circ W(\delta)_X.$$
		For this, we will call such $2$-cell just $\delta\:(f,\alpha)\aR{}(g,\beta)$. The identities are given by taking $\delta$ to be the appropriate identity $2$-cell. The horizontal composition is the one obtained by the description above; namely, given $\delta\:(f,\alpha)\aR{}(g,\beta)\:(A,X)\to(A',X')$ and $\varepsilon\:(f',\alpha')\aR{}(g',\beta')\:(A',X')\to(A'',X'')$ in $\Grothop{W}$,\v[-1]
		\begin{center}
			\linesep{1.5}
			\begin{tabular}{LL}
				\varepsilon\ast \delta = \left((\id{},\beta')\h \underline{\varepsilon}^{X'}\right)\ast \left((\id{},\beta)\h \underline{\delta}^X\right)=\left((\id{},\beta')\circ (\id{},W(g')(\beta))\right)\h \left(\underline{\varepsilon}^{W(g)(X)}\ast \underline{\delta}^X\right)=\\
				=(\id{},\beta'\circ W(g')(\beta))\h \underline{(\varepsilon\ast \delta)}^X
			\end{tabular}
		\end{center}
		and so it corresponds to the $2$-cell $\varepsilon\ast \delta$ in $\catfont{A}$. It is natural to define the vertical composition $(f,\alpha)\aR{\delta}(g,\beta)\aR{\delta'}(h,\gamma)$ in $\Grothop{W}$ to be the $2$-cell in $\Grothop{W}$ that corresponds to the $2$-cell $\delta'\circ\delta$ in $\catfont{A}$. And since we want $\groth{W}$ to be a $2$-functor, we need $\groth{W}$ to send a $2$-cell $(\id{},\beta)\h\underline{\delta}^X$ to the $2$-cell $\delta$ in $\catfont{A}$.

		It is then straightforward to show that we have given $\Grothop{W}$ the structure of a $2$-category and that $\groth{W}\:\Grothop{W}\to \catfont{A}$ is a $2$-functor. Moreover, we see that $\Grothop{W}$ is small if $\catfont{A}$ is small. We call $\groth{W}$ (or sometimes also just $\Grothop{W}$) the \dfn{$2$-$\catfont{Set}$-enriched Grothendieck construction} of $W\:\catfont{A}\to \catfont{CAT}$. Notice that we have also described the right notion of relaxed $2$-natural transformation that we need $\t{\varphi}$ to satisfy in order to encode the $2$-naturality of $\varphi$. It is a form of marked lax natural transformation, that is, a lax natural transformation such that certain structure $2$-cells are asked to be identities. And such slight strictness is necessary to encode the strict axioms of $2$-naturality of $\varphi$. This brings to \nameit{Definition}\ref{deflaxnormal} and to the new proof of the result that every weighted $2$-limit can be reduced to a \predfn{lax normal conical} one (for which we have given the ideas in this construction).
	\end{cons}

In short, we read from \nameit{Construction}\ref{2SetGrothconstrtoconicalize} the following explicit definition of the $2$-$\catfont{Set}$-enriched Grothendieck construction, that coincides with one of Street's~\cite{street_limitsindexedbycatvalued}.

\begin{defne}\label{defexpltwosetgroth}
	Let $F\:\catfont{B}\to\catfont{CAT}$ be a $2$-functor with $\catfont{B}$ a $2$-category. The \dfn{$2$-$\catfont{Set}$-enriched Grothendieck construction of $F$} is the $2$-functor $\groth{F}\:\Grothop{F}\to \catfont{B}$, given by the projection on the first component, with $\Grothop{F}$ such that:
	\begin{description}
		\item[an object of $\Grothop{F}$] is a pair $(B,X)$ with $B\in \catfont{B}$ and $X\in F(B)$;
		\item[a morphism $(B,X)\to (C,X')$ in $\Grothop{F}$] is a pair $(f,\alpha)$ with $f\:B\to C$ a morphism in $\catfont{B}$ and $\alpha\:F(f)(X)\to X'$ a morphism in $F(B)$;
		\item[a $2$-cell $(f,\alpha)\aR{}(g,\beta)\:(B,X)\to (C,X')$ in $\Grothop{W}$] is a $2$-cell $\delta\:f\aR{}g$ in $\catfont{B}$ such that $\alpha=\beta\circ F(\delta)_X$;
		\item[the compositions and identities] are as described in \nameit{Construction}\ref{2SetGrothconstrtoconicalize}.
	\end{description}
\end{defne}

	\begin{rem}\label{twosetgrothistrueusual}
		The $2$-$\catfont{Set}$-enriched Grothendieck construction is a natural extension of the usual Grothendieck construction. We believe that the latter should actually be conceived as the restriction of the former to $2$-presheaves into $\catfont{CAT}$ on a locally discrete $2$-category. After all, in dimension $1$ (see \nameit{Definition}\ref{defexplicgrothconstr}), the idea of reorganizing a family of sets as the fibres of a function with domain the disjoint union of the family was fully realized by the construction of the category of elements, admitting families of sets indexed by a category rather than by just a set. So we conceive the $2$-$\catfont{Set}$-enriched Grothendieck construction as the complete Grothendieck construction that lives in dimension $2$, rather than a particular case of a construction that lives in dimension $3$, and we will do the same with the notion of fibration that it produces (see Section~\ref{sectiontwosetgrothconstr}).

		It is interesting to notice that the $2$-$\catfont{Set}$-enriched Grothendieck construction and then also the usual Grothendieck construction are bound up with some form of laxness, being produced together with the relaxed notion of $2$-natural transformation that is necessary to encode the weighted $\catfont{Cat}$-(co)cylinders (\nameit{Construction}\ref{2SetGrothconstrtoconicalize}). We will explore this in more detail in Section~\ref{sectiontwosetgrothconstr}.
	\end{rem}

	\begin{defne}\label{deflaxnormal}
		Let $W\:\catfont{A}\to \catfont{CAT}$ be a $2$-functor with $\catfont{A}$ small, and consider $2$-functors $M,N\:\Grothop{W}\to \catfont{D}$. A \dfn{lax normal natural transformation $\alpha$ from $M$ to $N$}, denoted $\alpha\:M\alaxn{}N$,\v is a lax natural transformation $\alpha$ from $M$ to $N$ such that the structure $2$-cell on every morphism
		$$\p{f,\id{W(f)(X)}}\:(A,X)\to (B,W(f)(X))$$
		in $\Grothop{W}$ is the identity.
	\end{defne}

\begin{rem}\label{motivationlaxnormal}
	The lax normal natural transformations are a particular case of a more general notion of marked lax natural transformation introduced by Gray in~\cite{gray_formalcattheory}. Street showed in~\cite{street_limitsindexedbycatvalued} that this less general notion is totally sufficient to build all the general limits considered by Gray (showing that it is sufficient to build all the weighted limits and that all the limits considered by Gray are particular weighted ones).

	 Notice that, in some sense, the laxness of the lax normal natural transformations only belongs to the vertical part of $\Grothop{W}$. This idea will be understood even better after \nameit{Theorem}\ref{laxnormalareweighted} (the lax normal conical $2$-limits are weighted ones).
\end{rem}

	\begin{defne}\label{laxnormalconicallimits}
		Let $W\:\catfont{A}\to \catfont{CAT}$ be a $2$-functor with $\catfont{A}$ small, and let $F\:\Grothop{W} \to \catfont{C}$ be a $2$-functor. Notice that $\Grothop{W}$ is small, since $\catfont{A}$ is small. The \dfn{lax normal conical $2$-limit of $F$}, denoted as $\laxnlim{\Delta 1}{F}$, is (if it exists) an object $L\in \catfont{C}$ together with an isomorphism of categories
		$$\HomC{\catfont{C}}{U}{L}\cong \HomC{\mlaxn{\Grothopdiag{W}}{\catfont{CAT}}}{\Delta 1}{\HomC{\catfont{C}}{U}{F(-)}}$$
		$2$-natural in $U\in \catfont{C}\ensuremath{^{\operatorname{op}}}$,\v where $\mlaxn{\Grothop{W}}{\catfont{CAT}}$ is the $2$-category of $2$-functors, lax normal natural transformations and modifications from $\Grothop{W}$ to \catfont{CAT}. (Notice that indeed lax normal natural transformations compose well vertically.)

		\noindent When $\laxnlim{\Delta 1}{F}$ exists, taking $U=L$ and considering the identity on $L$ gives us in particular a lax normal natural transformation
		$$\lambda\:\Delta 1 \alaxn{}\HomC{\catfont{C}}{L}{F(-)},$$
		called the \dfn{universal lax normal cone}.

		For the definition of \predfn{lax normal conical $2$-colimit} in $\catfont{C}$, we just apply the definition of lax normal conical $2$-limit to a $2$-functor $F\:\Grothop{W}\to\catfont{C}\ensuremath{^{\operatorname{op}}}$. As usual, we prefer to consider instead $F\ensuremath{^{\operatorname{op}}}\:{\left(\Grothop{W}\right)}\ensuremath{^{\operatorname{op}}}\to\catfont{C}$, but this time we cannot rename ${\left(\Grothop{W}\right)}\ensuremath{^{\operatorname{op}}}$ as some $\Grothop{Z}$. So the definition we propose is the following.

		Let $W\:\catfont{A}\to\catfont{CAT}$ be a $2$-functor with $\catfont{A}$ small, and let $F\:{\left(\Grothop{W}\right)}\ensuremath{^{\operatorname{op}}}\to \catfont{C}$ be a $2$-functor. The \dfn{lax normal conical $2$-colimit of $F$}, denoted as $\laxncolim{\Delta 1}{F}$, is (if it exists) an object $C\in \catfont{C}$ together with an isomorphism of categories
		$$\HomC{\catfont{C}}{C}{U}\cong \HomC{\mlaxn{\Grothopdiag{W}}{\catfont{CAT}}}{\Delta 1}{\HomC{\catfont{C}}{F(-)}{U}}$$
		$2$-natural in $U\in \catfont{C}$.

		\noindent When $\laxncolim{\Delta 1}{F}$ exists we have, in place of $\lambda$, the \dfn{universal lax normal cocone}.
	\end{defne}

	\begin{rem}\label{fromgrothisnotrestrictive}
		Notice that considering just $2$-functors $F$ of the form $F\:\Grothop{W}\to\catfont{C}$ in \nameit{Definition}\ref{laxnormalconicallimits} is not restrictive at all. Indeed any $2$-category $\catfont{B}$ can be seen as the $2$-$\catfont{Set}$-enriched Grothendieck construction of the $2$-functor $\Delta 1\:\catfont{B}\to\catfont{CAT}$ constant at $\catfont{1}$. We have in fact that $\Grothop{\Delta 1}\cong \catfont{B}$ and that $\groth{\Delta 1}$ is the identity $2$-functor up to this isomorphism.

		It is important, though, to have a specified expression of the domain of the diagram $F$ in terms of a $2$-$\catfont{Set}$-enriched Grothendieck construction of some $2$-functor $W\:\catfont{A}\to\catfont{CAT}$ with $\catfont{A}$ small, as we need it in order to be able to say ``normal". The idea is that we want the laxness of the cones to only belong to the categories that form the family $W$ themselves rather than also to the connections between them (see also the proof of \nameit{Theorem}\ref{laxnormalareweighted}, that shows how the lax normal conical $2$-limits are weighted).
	\end{rem}

	\begin{exampl}\label{conicalarelaxnormalconical}
		Every conical $2$-limit is a lax normal conical one. Indeed, let $F\:\catfont{A}\to\catfont{C}$ be a $2$-functor. By \nameit{Remark}\ref{fromgrothisnotrestrictive}, we can view $\catfont{A}$ as the $2$-$\catfont{Set}$-enriched Grothendieck construction of the $2$-functor $\Delta 1\:\catfont{A}\to\catfont{CAT}$. And the lax normal natural transformations between $\catfont{A}=\Grothop{\Delta 1}$ and $\catfont{CAT}$ are just strict $2$-natural, as every morphism of $\Grothop{\Delta 1}$ is of the form $(f,\id{})$.
	\end{exampl}

	\begin{rem}
		We would now like to see that the lax normal conical $2$-limits are particular weighted $2$-limits. The proof we present (\nameit{Theorem}\ref{laxnormalareweighted}) does not seem to appear in the literature. There is a solution to this problem (without proof) in Street's paper~\cite{street_limitsindexedbycatvalued}, but that solution is actually about all the general $2$-limits introduced by Gray and is more complicated than the one presented here (the weight is given as the coidentifier of a certain $2$-cell with horizontal codomain the weight of lax conical limits). And since Street proved in~\cite{street_limitsindexedbycatvalued} that all the general $2$-limits introduced by Gray are built from the lax normal conical ones, we think a more explicit solution of the weight for lax normal conical $2$-limits could be valuable. It is also interesting to notice that the weight for lax normal conical $2$-limits (that is in the proof of \nameit{Theorem}\ref{laxnormalareweighted}) is much simpler than the weight for lax conical $2$-limits (that can be found in Street's~\cite{street_limitsindexedbycatvalued}). Indeed, the latter involves quotients of lax $2$-dimensional slices, while ordinary $1$-dimensional slices are enough for the former.
	\end{rem}

	\begin{teor}\label{laxnormalareweighted}
		Lax normal conical $2$-limits are particular weighted $2$-limits.
	\end{teor}
	\begin{proof}
		Let $Z\:\catfont{A}\to \catfont{CAT}$ be a $2$-functor with $\catfont{A}$ a small $2$-category, and let $F\:\Grothop{Z} \to \catfont{C}$ be a $2$-functor. We want to write the universal property of the lax normal conical $2$-limit of $F$ as the universal property of the $2$-limit of $F$ weighted by some $2$-functor $\operatorname{W}^{\laxn}\:\Grothop{Z}\to\catfont{CAT}$. It suffices to show that for every $2$-functor $N\:\catfont{C}\ensuremath{^{\operatorname{op}}}\to\m{\Grothop{Z}}{\catfont{CAT}}$, calling $N^U\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} N(U)$,  there is an isomorphism of categories
		\begin{equation}\label{laxnormalclas}
			\HomC{\mlaxn{\Grothopdiag{Z}}{\catfont{CAT}}}{\Delta 1}{N^U}\cong \HomC{\m{\Grothopdiag{Z}}{\catfont{CAT}}}{\operatorname{W}^{\laxn}}{N^U}
		\end{equation}
		$2$-natural in $U\in \catfont{C}\ensuremath{^{\operatorname{op}}}$. Let then $\varphi\:\Delta 1 \alaxn{} N^U$ be a lax normal natural transformation; we want to convert it into a $2$-natural transformation $[\varphi]\:\operatorname{W}^{\laxn}\aR{}N^U$.

		Given $(B,X')\in \Grothop{Z}$, we have a map $\varphi_{(B,X')}\:\catfont{1}\to N^U(B,X')$, i.e.\ an object $\varphi_{(B,X')}\in N^U(B,X')$. But we also have, for every morphism $(f,\alpha)\:(A,X)\to (B,X')$ in $\Grothop{Z}$, a $2$-cell
		\sq[l][6][7][\varphi_{(f,\alpha)}][2][2][0.5][l]{\catfont{1}}{{N^U}({A,X})}{\catfont{1}}{{N^U}({B,X'})}{\varphi_{(A,X)}}{}{{N^U}({f,\alpha})}{{\varphi}_{(B,X')}}
		i.e.\ a morphism $\varphi_{(f,\alpha)}$ in $N^U(B,X')$. So we want $\operatorname{W}^{\laxn}(B,X')$ to be some sort of slice that parametrizes all these data. But the idea behind lax normal natural transformations is that the laxness is all concentrated in the vertical part of $\Grothop{W}$, that is, on the morphisms of the form $(\id{A},\alpha)$ in $\Grothop{W}$ for some $A\in \catfont{A}$ and some morphism $\alpha\:X\to X'$ in $W(A)$. We can then parametrize just the vertical part of the data, defining
		\begin{fun}
			\operatorname{W}^{\laxn} & \: & \Grothop{Z} \hphantom{..}& \longrightarrow & \catfont{CAT} \\[1ex]
			&& \fib[n][3]{(B,X')}{(g,\beta)}{(C,X'')} &\mapsto & \fib[n][3]{\slice{Z(B)}{X'}}{\beta\circ Z(g)(-)}{\slice{Z(C)}{X''}}
		\end{fun}
		where the action of $\beta\circ Z(g)(-)$ on morphisms\v is given by $Z(g)(\operatorname{dom}(-))$. (This can be compared with the weight for lax conical $2$-limits, that considers quotients of the lax slices of the entire $\Grothop{Z}$ on $(B,X')$.)

		Given a $2$-cell $\delta\:(g,\beta)\aR{}(h,\gamma)\:{(B,X')}\to{(C,X'')}$, we define $\operatorname{W}^{\laxn}(\delta)$ to be the natural transformation\v[-2]
		\tc+[8]{\slice{Z(B)}{X'}}{\slice{Z(C)}{X''}}{\beta\circ Z(g)(-)}{\gamma \circ Z(h)(-)}{Z(\delta)_{\operatorname{dom}(-)}}
		Indeed, given $(X\ar{\alpha} X')\in \slice{Z(B)}{X'}$, we do have that $Z(\delta)_X$ is a morphism\v in $\slice{Z(C)}{X''}$ from $\beta\circ Z(g)(\alpha)$ to $\gamma\circ Z(h)(\alpha)$, as $Z(\delta)$ is a natural transformation and $\delta$ is a $2$-cell in $\Grothop{Z}$.\v And the naturality of $Z(\delta)_{\operatorname{dom}(-)}$ holds by naturality of $Z(\delta)$. It is then easy to show that $\operatorname{W}^{\laxn}$ is a $2$-functor.

		Now, we construct $[\varphi]\:\operatorname{W}^{\laxn}\aR{}N^U$ in the following way. Given $(B,X')\in \Grothop{Z}$, we set
		$$[\varphi]_{(B,X')}(\id{X'})\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \varphi_{(B,X')}$$
		whence we have to choose by naturality that $$[\varphi]_{(B,X')}(X\ar{\alpha}X')=N^U(\id{B},\alpha)\left(\varphi_{(B,X)}\right).$$
		And we can then set
		$$[\varphi]_{(B,X')}\left(\begin{cd}*[1.1][0.55]
			X\arrow[rd,"{\alpha}"']\arrow[rr,"{\alpha}"] \&\& X'\arrow[ld,equal] \\
			\& X'
		\end{cd}\right)\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \varphi_{(\id{B},\alpha)},$$
		whence we have to choose (looking again at the naturality of $[\varphi]$ that we want to obtain)
		$$[\varphi]_{(B,X')}\left(\begin{cd}*[1.2][0.6]
			S\arrow[rd,"{\theta\circ\alpha}"']\arrow[rr,"{\alpha}"] \&\& T\arrow[ld,"{\theta}"] \\
			\& X'
		\end{cd}\right)=N^U(\id{B},\theta)\left(\varphi_{(\id{B},\alpha)}\right).$$
		The component $[\varphi]_{(B,X')}$ is then a functor, thanks to the lax naturality of $\varphi$. And it is easy to show that $[\varphi]$ is a natural transformation, using that $\varphi$ is a lax normal natural transformation and how composition is defined in $\Grothop{Z}$. We show that $[\varphi]$ is $2$-natural. So take a $2$-cell $\delta\:{(g,\beta)}\aR{}{(h,\gamma)}\:{(B,X')}\to {(C,X'')}$ in $\Grothop{Z}$ and $(X\ar{\alpha}X')\in \slice{Z(B)}{X'}$; we prove that
		$$N^U(\id{C},\gamma\circ Z(h)(\alpha))\left(\varphi_{(\id{C},Z(\delta)_X)}\right)=N^U(\delta)_{N^U(\id{B},\alpha)\left(\varphi_{(B,X)}\right)}.$$
		But this is obtained combining the following two equations:
		$$\varphi_{(\id{C},Z(\delta)_X)}=\varphi_{(g,Z(\delta)_X)}=N^U\left(\underline{\delta}^X\right)_{\varphi_{(B,X)}}$$
		$$\delta\h(\id{B},\alpha)=(\id{C},\gamma\circ Z(h)(\alpha))\h\underline{\delta}^X.$$

		Consider now a modification $\Theta\:\varphi\aM{}\psi$ between two lax normal natural transformations $\varphi,\psi\:\Delta 1 \alaxn{} N^U$.\v We want to convert it into a modification $[\Theta]\:[\varphi]\aM{}[\psi]\:\operatorname{W}^{\laxn}\aR{}N^U$. Given $(B,X')\in \Grothop{Z}$, we define $[\Theta]_{(B,X')}$ to be the natural transformation with general component on $(\alpha\:X\to X')\in\slice{Z(B)}{X'}$ the morphism in $N^U(B,X')$
		$$[\Theta]_{(B,X'),\alpha}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} N^U(\id{B},\alpha)\left(\h\Theta_{(B,X)}\right)\:N^U(\id{B},\alpha)\left(\varphi_{(B,X)}\right)\to N^U(\id{B},\alpha)\left(\psi_{(B,X)}\right).$$
		Then $[\Theta]_{(B,X')}$ is natural since $\Theta$ is a modification, and $[\Theta]$ is a modification since $\Theta$ is a modification between lax normal natural transformations. Moreover this construction is certainly functorial.

		Conversely, starting from a $2$-natural transformation $\sigma\:\operatorname{W}^{\laxn}\aR{} N^U$, we convert it into a lax normal natural transformation $\ov{\sigma}\:\Delta 1 \alaxn{} N^U$. Given $(B,X')\in \Grothop{Z}$, we define $$\ov{\sigma}_{(B,X')}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \sigma_{(B,X')}(\id{X'}).$$
		And given a morphism $(f,\alpha)\:(A,X)\to (B,X')$ in $\Grothop{Z}$, we define
		$$\ov{\sigma}_{(f,\alpha)}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \sigma_{(B,X')}\left(\begin{cd}*[0.6][0.3]
			Z(f)(X)\arrow[rd,"{\alpha}"']\arrow[rr,"{\alpha}"] \&\&[0.8ex] X'\arrow[ld,equal] \\
			\& X'
		\end{cd}\right)$$
		noticing that $$\sigma_{(B,X')}(Z(f)(X)\ar{\alpha}X')=N^U(f,\alpha)\left(\sigma_{(A,X)}(\id{X})\right)=N^U(f,\alpha)\left(\ov{\sigma}_{(A,X)}\right)$$
		by naturality of $\sigma$. We then have that $\ov{\sigma}$ satisfies the $1$-dimensional part of being a lax normal natural transformation, by functoriality of the components of $\sigma$ and naturality of $\sigma$ (on morphisms). It remains to prove the $2$-dimensional part of $\ov{\sigma}$ being lax natural. So take a $2$-cell $\delta\:{(g,\beta)}\aR{}{(h,\gamma)}\:{(B,X')}\to {(C,X'')}$ in $\Grothop{Z}$; we want to show that\v[-3]
		\begin{equation*}
			\ov{\sigma}_{(h,\gamma)}\circ N^U(\delta)_{\h\ov{\sigma}_{(B,X')}}=\ov{\sigma}_{(g,\beta)}.
		\end{equation*}
		By the $2$-naturality of $\sigma$, we have that
		$$N^U(\delta)_{\h\ov{\sigma}_{(B,X')}}=N^U(\delta)_{\h{\sigma}_{(B,X')}(\id{X'})}=\sigma_{(C,X'')}\left(\begin{cd}*[0.6][0.3]
			Z(g)(X')\arrow[rd,"{\beta}"']\arrow[rr,"{Z(\delta)_{X'}}"] \&\& Z(h)(X')\arrow[ld,"{\gamma}"] \\
			\& X''
		\end{cd}\right)$$
		and we conclude by functoriality of $\sigma_{(C,X'')}$.

		Considering now\v a modification $\Xi\:\sigma\aM{}\rho\:\operatorname{W}^{\laxn} \aR{} N^U$, we convert it into a modification $\ov{\Xi}\:\ov{\sigma}\aM{}\ov{\rho}\:\Delta 1 \alaxn{} N^U$. Given $(B,X')\in \Grothop{Z}$, we define $$\ov{\Xi}_{(B,X')}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \Xi_{(B,X'),\id{X'}}.$$
		Then $\ov{\Xi}$ is a modification since $\Xi$ is a modification, and this construction is certainly functorial.

		It is now easy to see that the two constructions we have produced are inverses of each other, giving us an isomorphism of categories as in equation~\refs{laxnormalclas} for every $U\in \catfont{C}\ensuremath{^{\operatorname{op}}}$. And the $2$-naturality of such isomorphism holds trivially just by the fact that $N$ lands in $\m{\Grothop{Z}}{\catfont{CAT}}$.
	\end{proof}

	\begin{rem}
		The proof of \nameit{Theorem}\ref{laxnormalareweighted} works the same also for the lax normal conical $2$-colimit of a $2$-functor $F\:{\left(\Grothop{Z}\right)}\ensuremath{^{\operatorname{op}}}\to \catfont{C}$. Indeed the corresponding natural isomorphism of categories that we then want to prove is exactly the same we had in equation~\refs{laxnormalclas}. The only thing that changes is that $N$ becomes $N\:\catfont{C}\to\m{\Grothop{Z}}{\catfont{CAT}}$, but the naturality in $U$ of the isomorphism of categories in equation~\refs{laxnormalclas} continues to hold trivially just by the fact that $N$ lands in $\m{\Grothop{Z}}{\catfont{CAT}}$. We thus obtain the following corollary (of the proof of \nameit{Theorem}\ref{laxnormalareweighted}).
	\end{rem}

	\begin{coroll}\label{laxnormalcolimareweighted}
		Lax normal conical $2$-colimits are particular weighted $2$-colimits, and the weight that expresses them is $\operatorname{W}^{\laxn}$.
	\end{coroll}

	\begin{rem}
		We now present our new proof of the fact (firstly proved by Street in~\cite{street_limitsindexedbycatvalued}) that every weighted $2$-limit can be reduced to a lax normal conical one (and thus essentially conicalized). It will be based on the intuitive and elementary \nameit{Construction}\ref{2SetGrothconstrtoconicalize}.
	\end{rem}

	\begin{teor}\label{redlaxnormalconical}
		All the weighted $2$-limits can be reduced to lax normal conical ones. More precisely, given $2$-functors $F\:\catfont{A}\to \catfont{C}$ and $W\:\catfont{A}\to\catfont{CAT}$ with $\catfont{A}$ small, we have that
		$$\wlim{W}{F}\cong \laxnlim{\Delta 1}{\left(F\circ \groth{W}\right)}$$
		either side existing if the other does, where $\groth{W}$ is the $2$-$\catfont{Set}$-enriched Grothendieck construction of $W$ (see \nameit{Definition}\ref{defexpltwosetgroth}).
	\end{teor}
	\begin{proof}
		Looking at \nameit{Remark}\ref{needoflaxcones}, we first focus on essentially conicalizing the weighted $2$-colimits
		$$W\cong \wcolim{W}{\yy}$$
		with $W\:\catfont{A}\to \catfont{CAT}$ a $2$-presheaf with $\catfont{A}$ small and $\yy\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\m{\catfont{A}}{\catfont{CAT}}$ the $2$-Yoneda embedding, and then apply the lemma of continuity of a limit in its weight (\nameit{Lemma}\ref{continuityinweight}) to analogously essentially conicalize any weighted $2$-limit. Looking at \nameit{Construction}\ref{2SetGrothconstrtoconicalize}, given $W$ as above, we prove that there is an isomorphism of categories
		\begin{equation}\label{isoredlaxnormal}
			\scalebox{0.95}{$\HomC{\m{\catfont{A}}{\catfont{CAT}}}{W}{\HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{-}}{U}}\cong \HomC{\mlaxn{\Grothopdiag{W}}{\catfont{CAT}}}{\Delta 1}{\HomC{\m{\catfont{A}}{\catfont{CAT}}}{{\left(\yy\circ\h{\groth{W}}\ensuremath{^{\operatorname{op}}}\right)}(-)}{U}}$}
		\end{equation}
		$2$-natural in $U\in\m{\catfont{A}}{\catfont{CAT}}$, so that we can conclude that
		$$W\cong \wcolim{W}{\yy}\cong \laxncolim{\Delta 1}{\left(\yy\circ\h{\groth{W}}\ensuremath{^{\operatorname{op}}}\right)}.$$
		We have already described in \nameit{Construction}\ref{2SetGrothconstrtoconicalize} how to obtain from a $2$-natural transformation
		$$\varphi\:W\aR{}\HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{-}}{U};$$
		with $U\:\catfont{A}\to \catfont{CAT}$ a $2$-presheaf, a lax normal natural transformation
		$$\t{\varphi}\:\Delta 1 \alaxn{} \HomC{\m{\catfont{A}}{\catfont{CAT}}}{{\left(\yy\circ\h{\groth{W}}\ensuremath{^{\operatorname{op}}}\right)}(-)}{U}\:\Grothopdiag{W}\to\catfont{CAT}.$$
		Indeed, defining
		$$\t{\varphi}_{(A,X)}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \varphi_A(X)$$
		$$\t{\varphi}_{(f,\alpha)}=\varphi_B(\alpha)$$
		for every morphism $(f,\alpha)\:(A,X)\to (B,X')$ in $\Grothop{W}$, we have that $\t{\varphi}$ satisfies the $1$-dimensional part of being a lax normal natural transformation, since $\varphi$ is natural and the components of $\varphi$ are functors, and it also satisfies the $2$-dimensional part of being lax natural by the definition of the $2$-cells in $\Grothop{W}$ and the $2$-naturality of $\varphi$.

		Take now a modification
		$$\Theta\:\varphi\aM{}\psi\:W\aR{} \HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{-}}{U};$$
		we want to convert it into a modification between lax normal natural transformations
		$$\t{\Theta}\:\t{\varphi}\aM{}\t{\psi}\:\Delta 1 \alaxn{} \HomC{\m{\catfont{A}}{\catfont{CAT}}}{{\left(\yy\circ\h{\groth{W}}\ensuremath{^{\operatorname{op}}}\right)}(-)}{U}.$$
		Given $(A,X)\in \Grothop{W}$, we define
		$$\t{\Theta}_{(A,X)}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \Theta_{A,X}.$$
		$\t{\Theta}$ is then a modification since $\Theta$ is so and such assignment is certainly functorial. We now construct its inverse.

		Take a lax normal natural transformation
		$$\sigma\:\Delta 1 \alaxn{} \HomC{\m{\catfont{A}}{\catfont{CAT}}}{{\left(\yy\circ\h{\groth{W}}\ensuremath{^{\operatorname{op}}}\right)}(-)}{U}\:\Grothopdiag{W}\to\catfont{CAT};$$
		we want to convert it into a $2$-natural transformation
		$$\widehat{\sigma}\:W\aR{}\HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{-}}{U}.$$
		We define the component of $\widehat{\sigma}$ on $A\in \catfont{A}$ as
		\begin{fun}
			\widehat{\sigma}_{A} & \: & W(A) & \longrightarrow & \HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{A}}{U} \\[1ex]
			&& \fib[n][2.85]{X}{\alpha}{X'} & \mapsto & \fib[n][2.85]{\sigma_{(A,X)}}{\sigma_{(\id{A},\alpha)}}{\sigma_{(A,X')}}
		\end{fun}
		Then $\widehat{\sigma}_{A}$ is a functor, since $\sigma$ is lax natural. We prove that $\widehat{\sigma}$ is natural. So take $f\:A\to B$ a morphism in $\catfont{A}$. We need to show that the following square is commutative:
		\sq[5][7]{W(A)}{\HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{A}}{U}}{W(B)}{\HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{B}}{U}}{\widehat{\sigma}_A}{W(f)}{-\circ \y{f}}{\widehat{\sigma}_B}
		This holds by the fact that $\sigma$ is lax normal and that, given $\alpha\:X\to X'$ in $W(A)$, the following two morphisms are equal in $\Grothop{W}$:
		$$(A,X)\ar{(f,\id{W(f)(X)})}(B,W(f)(X))\ar{(\id{B},W(f)(\alpha))}(B,W(f)(X'))$$
		$$(A,X)\ar{(\id{A},\alpha)}(A,X')\ar{(f,\id{W(f)(X')})}(B,W(f)(X')).$$
		We now prove that $\widehat{\sigma}$ is $2$-natural. Given a $2$-cell $\delta\:f\aR{}g\:A\to B$, we need to show that for every $X\in W(A)$ we have
		$$\sigma_{(\id{B},W(\delta)_{X})}=\sigma_{(A,X)}\h\y{\delta}.\v$$
		But $\delta$ gives a $2$-cell $\underline{\delta}^X\:{(f,W(\delta)_X)}\aR{}{(g,\id{})}\:{(A,X)}\to{(B,W(g)(X))}$, whence by $\sigma$ being lax normal
		$$\sigma_{(A,X)}\h\y{\delta}=\sigma_{(f,W(\delta)_X)}=\sigma_{(\id{B},W(\delta)_X)}.$$

		Take now a modification between lax normal natural transformations
		$$\Xi\:\sigma\aM{}\rho\:\Delta 1 \alaxn{} \HomC{\m{\catfont{A}}{\catfont{CAT}}}{{\left(\yy\circ\h{\groth{W}}\ensuremath{^{\operatorname{op}}}\right)}(-)}{U};$$
		we want to convert it into a modification
		$$\widehat{\Xi}\:\widehat{\sigma}\aM{}\widehat{\rho}\:W\aR{} \HomC{\m{\catfont{A}}{\catfont{CAT}}}{\y{-}}{U}.$$
		We define the component of $\widehat{\Xi}$ on $A\in\catfont{A}$ to be the natural transformation with component on $X\in W(A)$
		$$\widehat{\Xi}_{A,X}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \Xi_{(A,X)}.$$
		We have that $\widehat{\Xi}$ is a modification since $\Xi$ is a modification between lax normal natural transformations. And this construction is certainly functorial.

		It is now easy to see that the two constructions we have produced are inverses of each other, giving us an isomorphism of categories as in equation~\refs{isoredlaxnormal}. And the $2$-naturality of such isomorphism of categories holds trivially. We have thus proved that every $2$-presheaf $W\:\catfont{A}\to\catfont{CAT}$ with $\catfont{A}$ small can be expressed as
		$$W\cong \laxncolim{\Delta 1}{\left(\yy\circ\h{\groth{W}}\ensuremath{^{\operatorname{op}}}\right)}.$$

		Consider now $2$-functors $F\:\catfont{A}\to\catfont{C}$ and $W\:\catfont{A}\to\catfont{CAT}$ with $\catfont{A}$ small. Then by the argument above and \nameit{Corollary}\ref{laxnormalcolimareweighted} we have
		$$W\cong \laxncolim{\Delta 1}{\left(\yy\circ\h{\groth{W}}\ensuremath{^{\operatorname{op}}}\right)}\cong \wcolim{\operatorname{W}^{\laxn}}{\left(\yy\circ\h{\groth{W}}\ensuremath{^{\operatorname{op}}}\right)}.$$
		By \nameit{Lemma}\ref{continuityinweight} and \nameit{Theorem}\ref{laxnormalareweighted}, we obtain
		$$\wlim{W}{F}\cong\wlim{\operatorname{W}^{\laxn}}{\left(\wlim{\left(\yy\circ\h {\groth{W}}\ensuremath{^{\operatorname{op}}}\right)(-)}{F}\right)}\cong\wlim{\operatorname{W}^{\laxn}}{\left(F\circ \groth{W}\right)}\cong\laxnlim{\Delta 1}{\left(F\circ \groth{W}\right)}$$
		either side existing if the other does, where the isomorphism in the middle is easy to prove.
	\end{proof}

	\begin{rem}\label{univlaxnormalcone}
		The proof of \nameit{Theorem}\ref{redlaxnormalconical}, together with \nameit{Lemma}\ref{continuityinweight} and the proofs of \nameit{Theorem}\ref{laxnormalareweighted} and \nameit{Corollary}\ref{laxnormalcolimareweighted}, also shows how to obtain the correspondence between the universal cylinder of a weighted $2$-limit and the universal lax normal cocone of the associated lax normal conical $2$-limit. Calling the two, respectively,
		$$\lambda\:W\aR{} \HomC{\catfont{C}}{L}{F(-)}$$
		$$\hats{\lambda}\:\Delta 1 \alaxn{} \HomC{\catfont{C}}{L}{\left(F\circ \groth{W}\right)(=)}$$
		for $F\:\catfont{A}\to\catfont{C}$ and $W\:\catfont{A}\to \catfont{CAT}$ two $2$-functors with $\catfont{A}$ small, the correspondence is given by the following equations, for every $(f,\alpha)\:(A,X)\to(B,X')$ in $\Grothop{W}$:
		\begin{equation}\label{lambdahats}
			\hats{\lambda}_{(A,X)}=\lambda_A(X)\quad\text{ and }\quad\hats{\lambda}_{(f,\alpha)}=\lambda_B(\alpha).
		\end{equation}
	\end{rem}

	\begin{prop}\label{preservereflect}
		A weighted $2$-limit is preserved or reflected precisely when its associated lax normal conical $2$-limit is so.
	\end{prop}
	\begin{proof}
		Clear after \nameit{Remark}\ref{univlaxnormalcone}.
	\end{proof}

	\begin{rem}\label{remoplaxnormalvslaxnormal}
		As weighted $2$-colimits in $\catfont{C}$ are just weighted $2$-limits in $\catfont{C}\ensuremath{^{\operatorname{op}}}$, we automatically obtain from \nameit{Theorem}\ref{redlaxnormalconical} the reduction of weighted $2$-colimits in $\catfont{C}$ to lax normal conical ones. More precisely, given $2$-functors $F\:\catfont{A}\to \catfont{C}$ and $W\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\catfont{CAT}$ with $\catfont{A}$ small, we obtain that
		$$\wcolim{W}{F}\cong \laxncolim{\Delta 1}{\left(F\circ {\groth{W}}\ensuremath{^{\operatorname{op}}}\right)},$$
		where $\groth{W}\:\Grothop{W}\to\catfont{A}\ensuremath{^{\operatorname{op}}}$.

		But notice that, when $W\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\catfont{CAT}$, there is a more natural Grothendieck construction we can do on $W$, i.e.\ the one which produces the projection on the first component $\h\groth{W}\:\Groth{W}\to\catfont{A}$, with $\Groth{W}$ defined as follows:
		\begin{description}
			\item[an object of $\Groth{W}$] is a pair $(A,X)$ with $A\in \catfont{A}$ and $X\in F(A)$;
			\item[a morphism $(A,X)\to (B,X')$ in $\Groth{W}$] is a pair $(f,\alpha)$ with $f\:A\to B$ a morphism in $\catfont{A}$ and $\alpha\:X\to W(f)(X')$ a morphism in $W(A)$;
			\item[a $2$-cell $(f,\alpha)\aR{}(g,\beta)\:(A,X)\to (B,X')$ in $\Groth{W}$] is a $2$-cell $\delta\:f\aR{}g$ in $\catfont{A}$ such that $W(\delta)_{X'}\circ \alpha=\beta$;
			\item[the compositions and identities] are analogous to the ones described in \nameit{Construction}\ref{2SetGrothconstrtoconicalize}.
		\end{description}
		In dimension $1$, given $Z\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\catfont{Set}$, we have that ${\left(\Grothop{Z}\right)}\ensuremath{^{\operatorname{op}}}$ and $\Groth{Z}$ coincide, but this is not true in dimension $2$.

		We can obtain a formula of reduction of the weighted $2$-colimits to some kind of conical ones that uses this more natural Grothendieck construction, changing ``lax" into ``oplax". Our idea, that does not seem to appear in the literature, is that it is more natural to reduce weighted $2$-limits to lax normal conical ones, but weighted $2$-colimits to \predfn{oplax normal conical} ones. And this brings to \nameit{Theorem}\ref{oplaxnormalareweighted} and \nameit{Theorem}\ref{redoplaxnormalconical}, which are original.
	\end{rem}

	\begin{defne}
		Let $W\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to \catfont{CAT}$ be a $2$-functor with $\catfont{A}$ small, and consider $2$-functors $M,N\:{\left(\Groth{W}\right)}\ensuremath{^{\operatorname{op}}}\to \catfont{D}$. An \dfn{oplax normal natural transformation $\alpha$ from $M$ to $N$}, denoted $\alpha\:M\aoplaxn{}N$,\v is an oplax natural transformation $\alpha$ from $M$ to $N$ such that the structure $2$-cell on every morphism
		$$\p{f,\id{W(f)(X)}}\:(A,W(f)(X))\al{} (B,X)$$
		in ${\left(\Groth{W}\right)}\ensuremath{^{\operatorname{op}}}$ is the identity.
	\end{defne}

	\begin{defne}\label{defoplaxnormalconical}
		Let $W\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to \catfont{CAT}$ be a $2$-functor with $\catfont{A}$ small, and let $F\:\Groth{W}\to \catfont{C}$ be a $2$-functor. Notice that $\Groth{W}$ is small, since $\catfont{A}$ is small. The \dfn{oplax normal conical $2$-colimit of $F$}, denoted as $\oplaxncolim{F}$, is (if it exists) an object $C\in \catfont{C}$ together with an isomorphism of categories
		$$\HomC{\catfont{C}}{C}{U}\cong \HomC{\moplaxn{{\left(\Grothdiag{W}\right)}\ensuremath{^{\operatorname{op}}}}{\catfont{CAT}}}{\Delta 1}{\HomC{\catfont{C}}{F(-)}{U}}$$
		$2$-natural in $U\in \catfont{C}$,\v where $\moplaxn{{\left(\Grothdiag{W}\right)}\ensuremath{^{\operatorname{op}}}}{\catfont{CAT}}$ is the $2$-category of $2$-functors, oplax normal natural transformations and modifications from ${\left(\Groth{W}\right)}\ensuremath{^{\operatorname{op}}}$ to \catfont{CAT}. (Notice that indeed oplax normal natural transformations compose well vertically.)

		\noindent When $\oplaxncolim{F}$ exists, taking $U=C$ and considering the identity on $C$ gives us in particular an oplax normal natural transformation
		$$\mu\:\Delta 1 \aoplaxn{}\HomC{\catfont{C}}{F(-)}{C},$$
		called the \dfn{universal oplax normal cocone}.
	\end{defne}

	\begin{teor}\label{oplaxnormalareweighted}
		Oplax normal conical $2$-colimits are particular weighted $2$-colimits. More precisely, given $2$-functors $Z\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\catfont{CAT}$ and $F\:\Groth{Z}\to\catfont{C}$ with $\catfont{A}$ small, the weight that realizes $\oplaxncolim{F}$ is
		\begin{fun}
			\operatorname{W}^{\oplaxn} & \: & {\left(\Groth{Z}\right)}\ensuremath{^{\operatorname{op}}} \hphantom{.}& \longrightarrow & \catfont{CAT} \\[1ex]
			&& \fib[o][2.85]{(B,X')}{(g,\beta)}{(C,X'')} &\mapsto & \fib[n][2.85]{\coslice{X'}{Z(B)}}{Z(g)(-)\circ\beta}{\coslice{X''}{Z(C)}}\\[6ex]
			&&\tcv*'[4.5]{(B,X')}{(C,X'')}{(g,\beta)}{(h,\gamma)}{\delta} & \mapsto &\h[-3] \tc*+[7]{\coslice{X'}{Z(B)}}{\coslice{X''}{Z(C)}}{\beta\circ Z(g)(-)}{\gamma \circ Z(h)(-)}{Z(\delta)_{\operatorname{cod}(-)}}\\
		\end{fun}
		where the action of $Z(g)(-)\circ\beta$ on morphisms\v is given by $Z(g)(\operatorname{cod}(-))$.
	\end{teor}
	\begin{proof}
		The proof is analogous to the one of \nameit{Theorem}\ref{laxnormalareweighted}.
	\end{proof}

	\begin{teor}\label{redoplaxnormalconical}
		All the weighted $2$-colimits can be reduced to oplax normal conical ones. More precisely, given $2$-functors $F\:\catfont{A}\to \catfont{C}$ and $W\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\catfont{CAT}$ with $\catfont{A}$ small, we have that
		$$\wcolim{W}{F}\cong \oplaxncolim{\left(F\circ \groth{W}\right)}$$
		either side existing if the other does, where $\groth{W}\:\Groth{W}\to\catfont{A}$ is the $2$-$\catfont{Set}$-enriched Grothendieck construction of $W$.
	\end{teor}
	\begin{proof}
		Exactly as in the proof of \nameit{Theorem}\ref{redlaxnormalconical}, we can prove that every $2$-presheaf $W\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\catfont{CAT}$ with $\catfont{A}$ small can be expressed as
		$$W\cong \wcolim{W}{\yy}\cong \oplaxncolim{\left(\yy\circ\h{\groth{W}}\right)}\cong \wcolim{\operatorname{W}^{\oplaxn}}{\left(\yy\circ\h{\groth{W}}\right)},$$
		where $\yy\:\catfont{A}\to\m{\catfont{A}\ensuremath{^{\operatorname{op}}}}{\catfont{CAT}}$ is the $2$-Yoneda embedding, and the last isomorphism is given by \nameit{Theorem}\ref{oplaxnormalareweighted}.

		Using \nameit{Lemma}\ref{continuityinweight}, we obtain
		$$\wcolim{W}{F}\cong \wcolim{\operatorname{W}^{\oplaxn}}{\left(\wcolim{\left(\yy\circ\h {\groth{W}}\right)(-)}{F}\right)}\cong \wcolim{\operatorname{W}^{\oplaxn}}{\left(F\circ \groth{W}\right)},$$
		either side existing if the other does, and we conclude by \nameit{Theorem}\ref{oplaxnormalareweighted}.
	\end{proof}

	\begin{rem}\label{univoplaxnormalcocone}
		Exactly as in \nameit{Remark}\ref{univlaxnormalcone}, in the notation of \nameit{Theorem}\ref{redoplaxnormalconical}, we can calculate the correspondence between the universal cocylinder of $\wcolim{W}{F}$ and the universal oplax normal cocone of $\oplaxncolim{\left(F\circ \groth{W}\right)}$. We find that it is the same as the one in equation~\refs{lambdahats}.
	\end{rem}

	\begin{prop}
		A weighted $2$-colimit is preserved or reflected precisely when its associated oplax normal conical $2$-colimit is so.
	\end{prop}
	\begin{proof}
		Clear after \nameit{Remark}\ref{univoplaxnormalcocone}.
	\end{proof}

	\begin{exampl}\label{univoplaxnormalcoconepresheaves}
		By the proof of \nameit{Theorem}\ref{redoplaxnormalconical}, every $2$-presheaf $W\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to\catfont{CAT}$ with $\catfont{A}$ small can be expressed as
		$$W\cong \oplaxncolim{\left(\yy\circ\h{\groth{W}}\right)}.$$
		The universal oplax normal cocone is given by
		$$\forall \fib[o][2.4]{(B,X')}{(f,\alpha)}{(A,X)} \text{ in } {\Grothdiag{W}} \quad\quad\quad{
			\begin{cd}*[4.5][2.25]
				\y{A} \arrow[rr,"{\ceil{X}}",""'{name=Q}] \arrow[d,"{\y{f}}"'] \&\& W\\
				\y{B} \arrow[rru,bend right,"{\ceil{X'}}"']\& \phantom{.} \arrow[Rightarrow,from=Q,"{\ceil{\alpha}}"{pos=0.36},shift right=1.5ex,shorten <=1ex,shorten >= 3.1ex]
		\end{cd}}$$

		In particular, taking $\catfont{A}=\catfont{1}$, we have that a $2$-presheaf on $\catfont{A}$ is just a small category $\catfont{D}$, and its $2$-$\catfont{Set}$-enriched Grothendieck construction just gives the functor to the terminal ${\catfont{D}}\to{\catfont{1}}$. We then obtain that every small category $\catfont{D}$ is the oplax normal conical $2$-colimit of $\Delta 1\:\catfont{D}\to \catfont{CAT}$, with oplax normal cocone given by
		$$\forall \fib[o]{D}{f}{C} \text{ in } \catfont{D} \quad\quad\quad {\begin{cd}*[4][2]
				\catfont{1} \arrow[rr,"C",""'{name=Q}] \arrow[d,equal] \&\& \catfont{D}\\
				\catfont{1} \arrow[rru,bend right,"D"']\& \phantom{.} \arrow[Rightarrow,from=Q,"{f}"{pos=0.355},shift right=1.5ex,shorten <=0.8ex,shorten >= 2.45ex]
		\end{cd}}$$
	\end{exampl}

	\begin{rem}\label{remoneisoplaxnormalconicaldense}
		\nameit{Example}\ref{univoplaxnormalcoconepresheaves} shows that $\catfont{1}$ is ``oplax normal conical dense" in \catfont{CAT}. And this may also serve as another motivation for the (op)lax normal conical $2$-(co)limits. Indeed, this is sufficient to express all the (co)powers as (op)lax normal conical $2$-(co)limits. And when $\catfont{C}$ is a (co)complete $2$-category, every $2$-(co)limit in $\catfont{C}$ can be written in terms of (co)powers and conical $2$-(co)limits. Remember also that by \nameit{Example}\ref{conicalarelaxnormalconical} every conical $2$-(co)limits is an (op)lax normal conical one. We can now understand how the (op)lax normal conical (co)limits are the result of the idea (anticipated at the beginning of this section) of considering the $2$-cells between $\catfont{1}$ and a category $\catfont{D}$ in order to capture the whole of $\catfont{D}$, rather than considering the functors from $\catfont{2}$ (that brings instead to the weighted $2$-limits).
	\end{rem}

	\section{The 2-$\catfont{Set}$-enriched Grothendieck construction}\label{sectiontwosetgrothconstr}

	In this section, we explore in detail the $2$-$\catfont{Set}$-enriched Grothendieck construction (\nameit{Definition}\ref{defexpltwosetgroth}), that firstly appeared in Street's~\cite{street_limitsindexedbycatvalued}, from a more abstract point of view, that is particularly relevant to, but not only, the (higher dimensional) elementary topos theory. This brings to the second main result of this paper (\nameit{Theorem}\ref{grothconstrislaxcomma}), which is expressing the $2$-$\catfont{Set}$-enriched Grothendieck construction as a \predfn{lax comma object in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$}, as defined here in \nameit{Definition}\ref{completeunivproplaxcomma} with a universal property that refines both the existing ones of Gray's~\cite{gray_formalcattheory} (unraveled by Kelly in~\cite{kelly_clubsanddoctrines}) and Lambert's~\cite{lambert_discretetwofib}. We explain how such laxness is needed to capture the $2$-$\catfont{Set}$-enriched Grothendieck construction (and the usual Grothendieck construction as well), and inscribe it in an original idea of $2$-$\catfont{V}$-enrichment, whence comes the name we gave to the studied construction.

	The lax $3$-category $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ of $2$-categories, $2$-functors, lax natural transformations and modifications is thus conceived as the archetypal $3$-dimensional elementary topos, suggesting to use the \predfn{lax comma objects in a lax $3$-category} (\nameit{Definition}\ref{completeunivproplaxcomma}) to regulate the classification process in a would-be \predfn{elementary $3$-topos}. Compare this with \nameit{Proposition}\ref{catofelementsiscomma}, that according to Weber's~\cite{weber_yonfromtwotop} presents $\catfont{CAT}$ as the archetypal elementary $2$-topos, with the construction of the category of elements as classification process, and suggests to regulate the classification by comma objects in dimension $2$. What gets classified in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ is the \predfn{discrete $2$-fibrations} with small fibres (a generalized version of the usual Grothendieck fibrations) introduced by Lambert in~\cite{lambert_discretetwofib}, that are a locally discrete version of Hermida's $2$-fibrations (defined in~\cite{hermida_somepropoffibasafibredtwocat}). Insisting on a $2$-$\catfont{Set}$-enrichment point of view, we will call them \predfn{$2$-$\catfont{Set}$-opfibrations}.

	We think that the $2$-$\catfont{Set}$-enriched Grothendieck construction should be taken into consideration towards the development of a general enriched Grothendieck construction (that considers general enriched fibrations). This will be explored in future work.

	\begin{rem}\label{nextstepislaxcomma}
		The guiding idea of this section is to categorify \nameit{Proposition}\ref{catofelementsiscomma}, presenting then the $2$-$\catfont{Set}$-enriched Grothendieck construction as the archetypal $3$-dimensional classification process (in the sense of a would-be tridimensional elementary topos theory). Analogously to the passage from pullbacks, that regulate the classification in dimension $1$, to comma objects, in dimension $2$, we expect to find now a further ``comma version" of the comma object that is suitable to a tridimensional ambient. The first instance of such process is given by \nameit{Proposition}\ref{laxnatinsidegrothconstr}, that shows the strictest filled square analogous to the one of \nameit{Proposition}\ref{catofelementsiscomma} we can have in the setting of the $2$-$\catfont{Set}$-enriched (but also just the usual) Grothendieck construction.
	\end{rem}

	\begin{prop}\label{laxnatinsidegrothconstr}
		Let $F\:\catfont{B}\to \catfont{CAT}$ be a $2$-functor and consider its $2$-$\catfont{Set}$-enriched Grothendieck construction. There is a lax normal natural transformation $\lambda$ of the form
		\begin{eqD*}
			\sq*[l][7][7][\laxn][2.7][2.2][0.52]{\Grothop{F}}{\catfont{1}}{\catfont{B}}{\catfont{CAT}}{}{\groth{F}}{\catfont{1}}{F}
		\end{eqD*}
	\end{prop}
	\begin{proof}
		Given $(A,X)\in \Grothop{F}$, we define the component of $\lambda$ on $(A,X)$ to be the functor $\catfont{1}\to F(A)$ corresponding to $X\in F(A)$. Given now a morphism $(f,\alpha)\:(A,X)\to (B,X')$ in $\Grothop{F}$, we define the structure $2$-cell of $\lambda$ on $(f,\alpha)$ to be the natural transformation corresponding to $\alpha$. It is straightforward to see that this is indeed a lax normal natural transformation, by construction of $\Grothop{F}$.
	\end{proof}

	\begin{rem}\label{needof2catlax}
		\nameit{Proposition}\ref{laxnatinsidegrothconstr} forces us to move out of $2\h[2]\mbox{-}\CAT$ in order to capture the $2$-$\catfont{Set}$-enriched (but also just the usual) Grothdendieck construction from an abstract point of view. Indeed, we need to at least admit the lax natural transformations as $2$-cells. And if we wish to recover the Grothendieck construction of pseudofunctors or of general lax functors into $\catfont{CAT}$, we also need to admit the lax functors as $1$-cells of our ambient. We will just consider strict $2$-functors for simplicity, to avoid at least the problems with the whiskering,	but we actually expect everything to hold for lax functors as well (despite not having investigated much of it yet).

		We call $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ the lax $3$-category of $2$-categories, $2$-functors, lax natural transformations and modifications. In his paper~\cite{lambert_discretetwofib}, Lambert has indeed proved that this forms a lax $3$-category, that is, a category enriched over the $1$-category of $2$-categories and normal lax functors.

		Be careful that $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ has no underlying $2$-category, since the interchange rule now only holds in a lax version, in the sense that we now have a modification between the two possible lax natural transformations. Indeed, consider two lax natural transformations
		\begin{cd}
			\catfont{A}\arrow[r,bend left,"{F}",""'{name=A}]\arrow[r,bend right,"{G}"',""{name=B}]\&\catfont{B} \arrow[r,bend left,"{H}",""'{name=C}]\arrow[r,bend right,"{K}"',""{name=D}]\&\catfont{C};
			\arrow[from=A,to=B,Rightarrow,"{\alpha}"]
			\arrow[from=C,to=D,Rightarrow,"{\beta}"]
		\end{cd}
		Then for every $A\in \catfont{A}$, the component $\alpha_A$ is a morphism $F(A)\ar{}G(A)$ in $\catfont{B}$ and we can consider the structure $2$-cell of $\beta$ on such morphism. We obtain that $\beta_{\alpha_A}$ is a $2$-cell in $\catfont{C}$ of the form
		\begin{cd}
			H(F(A))\arrow[r,"{\beta_{F(A)}}"]\arrow[d,"{H(\alpha_A)}"']\& K(F(A))\arrow[d,"{K(\alpha_A)}"]\arrow[ld,Rightarrow,"{\beta_{\alpha_{A}}}",shorten <=2.7ex,shorten >=2.2ex]\\
			H(G(A))\arrow[r,"{\beta_{G(A)}}"']\& K(G(A))
		\end{cd}
		And the $\beta_{\alpha_A}$'s collect into a modification $\beta_\alpha$, since the axiom we should check on a morphism $f\:A\to A'$ in $\catfont{A}$ is given by the $2$-dimensional property of $\beta$ being a lax natural transformation, applied to the $2$-cell $\alpha_f$ in $\catfont{B}$.
	\end{rem}

	\begin{rem}
		\nameit{Remark}\ref{nextstepislaxcomma} and \nameit{Proposition}\ref{laxnatinsidegrothconstr} lead us to use the concept of \predfn{lax comma object}. This concept appears and is heavily used in Gray's book~\cite{gray_formalcattheory}, but without giving a complete universal property suitable to the lax $3$-categorical ambient $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$. Kelly just unraveled in~\cite{kelly_clubsanddoctrines} the partial universal property presented by Gray, while Lambert attempted in~\cite{lambert_discretetwofib} to give a better one but missing the uniqueness result in the $2$-dimensional part and giving only a partial $3$-dimensional part. We give in \nameit{Definition}\ref{completeunivproplaxcomma} (see also \nameit{Proposition}\ref{laxcommaisobject}) the complete universal property of the lax comma object, that refines both the ones of Gray (and Kelly) and Lambert.

		In order to distinguish the explicit definition (given in Gray's~\cite{gray_formalcattheory}) from the complete universal property of the lax comma object, we will call the former ``\predfn{lax comma}" and the latter ``\predfn{lax comma object in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$}". However, we will use the same symbol for both, justified by \nameit{Proposition}\ref{laxcommaisobject}.
	\end{rem}

	\begin{defne}
		Let $F\:\catfont{A}\to \catfont{C}$ and $G\:\catfont{B}\to \catfont{C}$ be $2$-functors. The \dfn{lax comma from $F$ to $G$} is the $2$-category $\laxcomma{F}{G}$ that is given by the following data:
		\begin{description}
			\item[an object of $\laxcomma{F}{G}$] is a triple $(A,B,h)$ with $A\in \catfont{A}$, $B\in \catfont{B}$ and $h\:F(A)\to G(B)$ a morphism in $\catfont{C}$;
			\item[a $1$-cell $(A,B,h)\to (A',B',h')$ in $\laxcomma{F}{G}$] is a triple $(f,g,\varphi)$ with $f\:A\to A'$ in $\catfont{A}$, $g\:B\to B'$ in $\catfont{B}$ and
			\sq[l][7][7][\varphi][2.3][2.3]{F(A)}{G(B)}{F(A')}{G(B')}{h}{F(f)}{G(g)}{h'}
			a $2$-cell in $\catfont{C}$;
			\item[a $2$-cell $(f,g,\varphi)\aR{}(f',g',\varphi')\:(A,B,h)\to (A',B',h')$] is a pair $(\alpha,\beta)$ with $\alpha\:f\aR{}f'$ in $\catfont{A}$ and $\beta\:g\aR{}g'$ in $\catfont{B}$ such that
			\twonats[1.95][8][7][1.05][3.45][2]{F(A)}{G(B)}{F(A')}{G(B')}{F(f')}{F(f)}{G(g')}{G(g)}{h}{h'}{F(\alpha)}{G(\beta)}{\varphi}{\varphi'}
			\item[the composition] of $1$-cells is given by pasting and that of $2$-cells is inherited by the ones in $\catfont{A}$ and $\catfont{B}$.
		\end{description}

		The \dfn{oplax comma from $F$ to $G$} is the co of the lax comma from $F\ensuremath{^{\operatorname{co}}}$ to $G\ensuremath{^{\operatorname{co}}}$.
	\end{defne}

	The following proposition shows the partial universal property of the lax comma object presented by Gray in~\cite{gray_formalcattheory}.

	\begin{prop}\label{univproplaxcommagray}
		Let $F\:\catfont{A}\to \catfont{C}$ and $G\:\catfont{B}\to \catfont{C}$ be $2$-functors. The lax comma from $F$ to $G$ is equivalently given by the enriched conical limit in $2\h[2]\mbox{-}\CAT$ of the diagram
		\begin{cd}[4][4]
			\catfont{A} \arrow[rd,"{F}"']\&\& \ARoplax{\catfont{C}} \arrow[rd,"{\operatorname{cod}}"] \arrow[ld,"{\operatorname{dom}}"']\&\& \catfont{B}\arrow[ld,"{G}"]\\
			\& \catfont{C} \&\& \catfont{C}
		\end{cd}
		where $\ARoplax{\catfont{C}}$ is the lax comma from $\Id{\catfont{C}}$ to $\Id{\catfont{C}}$ \pteor{in some sense, the fundamental one}.
	\end{prop}

	\begin{rem}
		But there is a better universal property that the lax comma satisfies. Indeed, it is a \predfn{lax comma object in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$}, as defined here below.
	\end{rem}

	\begin{defne}\label{completeunivproplaxcomma}
		Let $\catfont{Q}$ be a lax $3$-category and consider $1$-cells $F\:\catfont{A}\to \catfont{C}$ and $G\:\catfont{B}\to \catfont{C}$ in $\catfont{Q}$. The \dfn{lax comma object in $\catfont{Q}$ from $F$ to $G$} is, if it exists, an object $\laxcomma{F}{G}\in \catfont{Q}$ together with a $2$-cell
		\begin{cd}
			\laxcomma{F}{G} \arrow[r,"{\partial_0}"]\arrow[d,"{\partial_1}"']\& \catfont{A}\arrow[d,"{F}"]\arrow[ld,Rightarrow,"{\lambda}",shorten <= 2.7ex, shorten >=2.2ex]\\
			\catfont{B}\arrow[r,"{G}"']\& \catfont{C}
		\end{cd}
		in $\catfont{Q}$ that is universal in the following lax $3$-categorical sense:
		\begin{enum}
			\item for every $2$-cell $\gamma\:F\circ P\aR{}G\circ Q\:\catfont{M}\to \catfont{C}$, there exists a unique $1$-cell $V\:\catfont{M}\to \laxcomma{F}{G}$ such that
			\begin{eqD*}
				\begin{cd}*[6.5][6.5]
					\catfont{M}\arrow[rrd,bend left,"{P}"]\arrow[rdd,bend right,"{Q}"']\\[-5ex]
					\&[-5ex]\& \catfont{A}\arrow[d,"{F}"]\arrow[ld,Rightarrow,shift right=3.4ex,"{\gamma}"{pos=0.53},shorten <= 2.2ex, shorten >=1.4ex]\\
					\&\catfont{B}\arrow[r,"{G}"']\& \catfont{C}
				\end{cd}\quad = \quad
				\begin{cd}*[6.5][6.5]
					\catfont{M}\arrow[rrd,bend left,"{P}"]\arrow[rdd,bend right,"{Q}"']\arrow[rd,"{V}",dashed,shorten <=-0.2ex,shorten >=-0.2ex]\\[-5ex]
					\&[-5ex]\laxcomma{F}{G} \arrow[r,"{\partial_0}"]\arrow[d,"{\partial_1}"']\& \catfont{A}\arrow[d,"{F}"]\arrow[ld,Rightarrow,"{\lambda}",shorten <= 2.7ex, shorten >=2.2ex]\\
					\&\catfont{B}\arrow[r,"{G}"']\& \catfont{C}
				\end{cd}
			\end{eqD*}
			\item for every $1$-cells $V,W\:\catfont{M}\to \laxcomma{F}{G}$ and every $3$-cell
			\begin{eqD*}
				\begin{cd}*
					\catfont{M}\arrow[dd,"{W}"']\arrow[rd,"{V}"]\\[-5ex]
					\&[-4ex]\laxcomma{F}{G}\arrow[ld,Rightarrow,"{\Delta}",shorten <=0ex, shorten >=0ex] \arrow[r,"{\partial_0}"]\arrow[dd,"{\partial_1}"']\& \catfont{A}\arrow[dd,"{F}"]\arrow[ldd,Rightarrow,"{\lambda}",shorten <= 2.7ex, shorten >=2.2ex]\\[-5ex]
					\laxcomma{F}{G}\arrow[rd,"{\partial_1}"']\\[-5ex]
					\&\catfont{B}\arrow[r,"{G}"']\& \catfont{C}
				\end{cd} \quad\aMM{\Xi}\quad
				\begin{cd}*
					\catfont{M}\arrow[rr,"{V}"]\arrow[rd,"{W}"']\&[-5ex]\&[-7ex] \laxcomma{F}{G}\arrow[rd,"{\partial_0}"]\arrow[ld,Rightarrow,"{\Gamma}"]\\[-4ex]
					\&\laxcomma{F}{G} \arrow[rr,"{\partial_0}"]\arrow[d,"{\partial_1}"']\&\&[-5ex]\catfont{A}\arrow[d,"{F}"]\arrow[lld,Rightarrow,"{\lambda}",shorten <= 2.7ex, shorten >=2.2ex]\\
					\&\catfont{B}\arrow[rr,"{G}"']\&\& \catfont{C}
				\end{cd}
			\end{eqD*}
			for $2$-cells $\Gamma$ and $\Delta$, there exists a unique $2$-cell $\nu\:V\aR{}W$ such that
			$$\partial_0\h \nu=\Gamma, \quad \partial_1\h \nu=\Delta, \quad \lambda_{\nu}=\Xi;$$
			notice that we are precisely asking that $\Xi$ corresponds to the $3$-cell given by the lax interchange rule in $\catfont{Q}$ of
			\begin{cd}
				\catfont{M}\arrow[r,bend left,"{V}"{pos=0.57},""'{name=A}]\arrow[r,bend right,"{W}"'{pos=0.57},""{name=B}]\&\laxcomma{F}{G} \arrow[r,bend left,"{F\circ \partial_0}"{pos=0.43},""'{name=C}]\arrow[r,bend right,"{G\circ \partial_1}"'{pos=0.43},""{name=D}]\&\catfont{C};
				\arrow[from=A,to=B,Rightarrow,"{\nu}"]
				\arrow[from=C,to=D,Rightarrow,"{\lambda}"]
			\end{cd}
			\item for every $2$-cells $\nu,\omega\:V\aR{}W\:\catfont{M}\to \laxcomma{F}{G}$ and every pair of $3$-cells $\Phi\:\partial_0\h \nu\aM{}\partial_0\h \omega$ and $\Psi\:\partial_1\h \nu\aM{}\partial_1\h \omega$ such that
			\begin{eqD}{tridimpartcompleteunivproplaxcomma}
				\lambda_{\omega}\circ
				\begin{cd}*
					\catfont{M}\arrow[dd,"{W}"']\arrow[rd,"{V}"]\\[-5ex]
					\&[-4ex]\laxcomma{F}{G}\arrow[ld,Rightarrow,bend right,"{\partial_1\h \nu}"'{pos=0.42},""'{name=A},shorten <=0.4ex, shorten >=0.4ex]\arrow[ld,Rightarrow,bend left,"{\partial_1\h \omega}"{pos=0.58},""{name=B},shorten <=0.4ex, shorten >=0.4ex] \arrow[r,"{\partial_0}"]\arrow[dd,"{\partial_1}"]\& \catfont{A}\arrow[dd,"{F}"]\arrow[ldd,Rightarrow,"{\lambda}",shorten <= 2.7ex, shorten >=2.2ex]\\[-5ex]
					\laxcomma{F}{G}\arrow[rd,"{\partial_1}"']\\[-5ex]
					\&\catfont{B}\arrow[r,"{G}"']\& \catfont{C}
					\arrow[from=A,to=B,triple,"{\Psi}",shorten <=1.1ex,shorten >=0.9ex]
				\end{cd} \quad=\quad
				\begin{cd}*
					\catfont{M}\arrow[rr,"{V}"]\arrow[rd,"{W}"']\&[-5ex]\&[-7ex] \laxcomma{F}{G}\arrow[rd,"{\partial_0}"]\arrow[ld,Rightarrow,bend right,"{\partial_0\h \nu}"'{pos=0.58},""'{name=C},shorten <=0.4ex,shorten >=0.4ex]\arrow[ld,Rightarrow,bend left,"{\partial_0\h\omega}"{pos=0.42},""{name=D},shorten <=0.4ex,shorten >=0.4ex]\\[-4ex]
					\&\laxcomma{F}{G} \arrow[rr,"{\partial_0}"']\arrow[d,"{\partial_1}"']\&\&[-5ex]\catfont{A}\arrow[d,"{F}"]\arrow[lld,Rightarrow,"{\lambda}",shorten <= 2.7ex, shorten >=2.2ex]\\
					\&\catfont{B}\arrow[rr,"{G}"']\&\& \catfont{C}
					\arrow[from=C,to=D,triple,"{\Phi}",shorten <=1.1ex,shorten >=0.9ex]
				\end{cd}\circ \lambda_{\nu}
			\end{eqD}
			there exists a unique $3$-cell $\Theta\:\nu\aM{}\omega$ such that $\partial_0\h \Theta=\Phi$ and $\partial_1\h \Theta=\Psi$.
		\end{enum}
	\end{defne}

	\begin{prop}\label{laxcommaisobject}
		Let $F\:\catfont{A}\to \catfont{C}$ and $G\:\catfont{B}\to \catfont{C}$ be $2$-functors. Then there is a lax natural transformation
		\begin{cd}
			\laxcomma{F}{G} \arrow[r,"{\partial_0}"]\arrow[d,"{\partial_1}"']\& \catfont{A}\arrow[d,"{F}"]\arrow[ld,Rightarrow,"{\lambda}",shorten <= 2.7ex, shorten >=2.2ex]\\
			\catfont{B}\arrow[r,"{G}"']\& \catfont{C}
		\end{cd}
		that exhibits the lax comma $\laxcomma{F}{G}$ as the lax comma object in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ from $F$ to $G$. Moreover, in condition $(ii)$ of lax comma object in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$, if $\Gamma$ and $\Delta$ \pteor{that can be general lax natural transformations} are both strict $2$-natural \pteor{resp. pseudo-natural} then also $\nu$ is so.
	\end{prop}
	\begin{proof}
		Firstly, we construct $\lambda$. Given $(A,B,h)\in \laxcomma{F}{G}$, we define the component of $\lambda$ on it to be $h$. Given a morphism $(f,g,\varphi)\:(A,B,h)\to (A',B',h')$ in $\laxcomma{F}{G}$, we define the structure $2$-cell of $\lambda$ on it to be $\varphi$. It is then straightforward to show that $\lambda$ is a lax natural transformation.

		For condition $(i)$ of lax comma object in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$, since $\lambda$ picks the third component of objects and morphisms in $\laxcomma{F}{G}$ and the other two components are determined by the projections through $\partial_0$ and $\partial_1$, we have to define $V$ as
		$$V(M)\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} (P(M),Q(M),\gamma_M)$$
		$$V(m)\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} (P(m),Q(m),\gamma_m)$$
		for every $M\in \catfont{M}$ and every morphism $m$ in $\catfont{M}$. This is easily checked to be a $2$-functor, and it works by construction.

		For $(ii)$, take an arbitrary $\Xi$ as above. We see that the three requests
		$$\partial_0\h \nu=\Gamma, \quad \partial_1\h \nu=\Delta, \quad \lambda_{\nu}=\Xi$$
		force us to construct the component of $\nu$ on an arbitrary $M\in \catfont{M}$ to be the morphism $(\Gamma_M,\Delta_M,\Xi_M)$ in $\laxcomma{F}{G}$. Then the structure $2$-cell of $\nu$ on a morphism $m\:M\to M'$ in $\catfont{M}$, being a $2$-cell in $\laxcomma{F}{G}$, is determined by its projections through $\partial_0$ and $\partial_1$. So we are forced to define $\nu_m$ to be the $2$-cell $(\Gamma_m,\Delta_m)$ in $\laxcomma{F}{G}$. This is indeed a $2$-cell since $\Xi$ is a modification. It is straightforward to check that $\nu$ is a lax natural transformation, since $\Gamma$ and $\Delta$ are so. And we immediatly see that if both $\Gamma$ and $\Delta$ are strict $2$-natural (resp. pseudo-natural) then also $\nu$ is so. The observation that $\nu$ is then the unique lax natural transformation $V\alax{}W$ such that the modification corresponding to the lax interchange rule in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ of $\nu$ and $\lambda$ coincides with $\Xi$ follows from \nameit{Remark}\ref{needof2catlax}.

		For $(iii)$, let $M\in \catfont{M}$. Since the component $\Theta_M$ will be a $2$-cell in $\laxcomma{F}{G}$, it is determined by its projections through $\partial_0$ and $\partial_1$. So we need to define
		$$\Theta_M\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} (\Phi_M,\Psi_M).$$
		That this is indeed a $2$-cell $\nu_M\aR{}\omega_M$ in $\laxcomma{F}{G}$ is guaranteed by equation~\refs{tridimpartcompleteunivproplaxcomma}, taking components on $M$. The condition that the $\Theta_M$'s need to satisfy in order for them to collect into a modification $\Theta$ is then an equality between $2$-cells in $\laxcomma{F}{G}$, and thus it suffices to check its projections through $\partial_0$ and $\partial_1$. But those two resulting conditions are given by the fact that both $\Phi$ and $\Psi$ are modifications.
	\end{proof}

	\begin{rem}
		Notice from \nameit{Definition}\ref{completeunivproplaxcomma} that the lax comma object in a lax $3$-category really is an upgrade of the comma object to a lax $3$-dimensional ambient. Indeed, a lax comma object in a $2$-category is precisely a comma object, since any $\Xi$ of \nameit{Definition}\ref{completeunivproplaxcomma} is then forced to be the identity, and the tridimensional part becomes trivial. Interestingly, the uniqueness in the $2$-dimensional part of the universal property of the lax comma object in a lax $3$-category is obtained by considering the lax interchange rule.

		The universal property of \nameit{Proposition}\ref{univproplaxcommagray} is obtained precisely by restricting ourselves to consider as $\Gamma$ and $\Delta$ only strict $2$-natural transformations.
	\end{rem}

	\begin{teor}\label{grothconstrislaxcomma}
		Let $F\:\catfont{B}\to \catfont{CAT}$ be a $2$-functor. The $2$-$\catfont{Set}$-enriched Grothendieck construction, defined explicitly in \nameit{Construction}\ref{2SetGrothconstrtoconicalize}, is equivalently given by the lax comma object
		\begin{eqD}{grothconstrislaxcommadiagram}
			\sq*[l][7][7][\lax \opn{comma}][2.7][2.2][0.65]{\Grothop{F}}{\catfont{1}}{\catfont{B}}{\catfont{CAT}}{}{\groth{F}}{\catfont{1}}{F}
		\end{eqD}
		in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ and such lax comma object is presented by the lax normal natural transformation $\lambda$ of \nameit{Proposition}\ref{laxnatinsidegrothconstr}.

		As a consequence, it is then also given by the strict $3$-pullback in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ between $F$ and the replacement $\tau$ of $\catfont{1}\:\catfont{1}\to \catfont{CAT}$ obtained by taking the lax comma object of $\catfont{1}\:\catfont{1}\to \catfont{CAT}$ along the identity of $\catfont{CAT}$ \pteor{that is a lax $3$-dimensional version of the lax limit of the arrow $\catfont{1}\:\catfont{1}\to \catfont{CAT}$}:
		\begin{eqD*}
			\begin{cd}*
				\Grothop{F}\PB{rd} \arrow[d,"\groth{F}"'] \arrow[r] \& \catfont{CAT}_{\bullet,\operatorname{lax}} \arrow[d,"{\tau}"'] \arrow[r]\& \catfont{1} \arrow[d,"{\catfont{1}}"]\arrow[ld,Rightarrow,shorten <=2.7ex,shorten >=2.2ex,"\lax \opn{comma}"{pos=0.65}] \\
				\catfont{B}\arrow[r,"F"'] \& \catfont{CAT} \arrow[r,equal]\& \catfont{CAT}
			\end{cd}
		\end{eqD*}
		The domain of $\tau$ is a lax pointed version of $\catfont{CAT}$, whence the notation $\catfont{CAT}_{\bullet,\operatorname{lax}}$.
	\end{teor}
	\begin{proof}
		The proof is a straightforward calculation. The fact that $\groth{F}$ is then also the strict $3$-pullback of $\tau$ is readily checked by showing that such strict $3$-pullback satisfies the universal property of the lax comma object $\laxcomma{\catfont{1}}{F}$ in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ that we have presented in \nameit{Definition}\ref{completeunivproplaxcomma}, using the universal properties of $\laxcomma{\catfont{1}}{\Id{\catfont{CAT}}}$ and of the strict $3$-pullback together with some basics of the calculus of pasting.
	\end{proof}

	\begin{rem}
		Before commenting on \nameit{Theorem}\ref{grothconstrislaxcomma} (we will in \nameit{Remark}\ref{remarchetypalthreedimclassifproc}), we use it to canonically extend the $2$-$\catfont{Set}$-enriched Grothendieck construction by functoriality.
	\end{rem}

	\begin{prop}\label{twosetgrothcanonicallyextendstwofunctor}
		By \nameit{Theorem}\ref{grothconstrislaxcomma}, the $2$-$\catfont{Set}$-enriched Grothendieck construction canonically extends to a $2$-functor
		$$\groth{-}\:\m{\catfont{B}}{\catfont{CAT}}\to \slice{2\h[2]\mbox{-}\CAT}{\catfont{B}}$$
		and to one with domain $\mlax{\catfont{B}}{\catfont{CAT}}$ as well, for every $2$-category $\catfont{B}$.
	\end{prop}
	\begin{proof}
		Given a $2$-natural transformation $\varphi\:F\aR{}G\:\catfont{B}\to \catfont{CAT}$ - also a lax natural transformation would equally work - we define $\groth{\varphi}$ as the unique morphism $\groth{\varphi}\:\Grothop{F}\to \Grothop{G}$ induced by the universal property of the lax comma object $\Grothop{G}$ in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ applied to the lax natural transformation
		\begin{cd}[7][8]
			{\Grothop{F}}\arrow[d,"{\groth{F}}"'] \arrow[r,"{}"]\& {\catfont{1}}\arrow[ld,Rightarrow,shift right=1ex,"{\lambda^F}"'{pos=0.475, inner sep=0.25ex},shorten <=3.2ex, shorten >=3.3ex] \arrow[d,"{\catfont{1}}"]\\
			{\catfont{B}} \arrow[r,bend left,"{F}"{pos=0.64},""'{name=A}]\arrow[r,bend right,"{G}"',""{name=B}] \& {\catfont{CAT}}
			\arrow[Rightarrow,from=A,to=B,shorten <=0.4ex,shorten >=0.4ex,"\varphi"]
		\end{cd}
		where $\lambda^F$ is the lax natural transformation that presents $\Grothop{F}$ as a lax comma object in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$. Explicitly, for every $2$-cell $\delta\:(f,\alpha)\aR{}(g,\beta)\:(B,X)\to (C,X')$ in $\Grothop{F}$
		$$\groth{\varphi}(B,X)=(B,\varphi_B(X)) \quad \text{and} \quad \groth{\varphi}(f,\alpha)=(f,\varphi_C(\alpha))\quad \text{and} \quad \groth{\varphi}(\delta)=\delta.$$

		Given a modification $\Theta\:\varphi\aM{} \psi\:F\aR{}G\:\catfont{B}\to \catfont{CAT}$, we define $\groth{\Theta}$ as the unique $2$-natural transformation induced by the universal property of the lax comma object $\Grothop{G}$ in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ applied, in the notation of \nameit{Definition}\ref{completeunivproplaxcomma} to $V=\groth{\varphi}$, $W=\groth{\psi}$, $\Gamma=\id{}$, $\Delta=\id{}$ and $\Xi$ given by
		\begin{cd}[7][8]
			{\Grothop{F}}\arrow[d,"{\groth{F}}"'] \arrow[r,"{}"]\& {\catfont{1}}\arrow[ld,Rightarrow,shift right=1ex,"{\lambda^F}"'{pos=0.475},shorten <=3.2ex, shorten >=3.3ex] \arrow[d,"{\catfont{1}}"]\\
			{\catfont{B}} \arrow[r,bend left=35,"{F}"{pos=0.64},""'{name=A}]\arrow[r,bend right=35,"{G}"',""{name=B}] \& {\catfont{CAT}}
			\arrow[Rightarrow,from=A,to=B,bend right=40,shift right=1ex,shorten <=0.4ex,shorten >=0.4ex,"\varphi"',""{name=L}]
			\arrow[Rightarrow,from=A,to=B,bend left=40,shift left=1ex,shorten <=0.4ex,shorten >=0.4ex,"\psi",""'{name=R}]
			\arrow[triple,from=L,to=R,shorten <=0ex,shorten >=-0.2ex,"\Theta"{inner sep=1ex}]
		\end{cd}
		Explicitly, the component of $\groth{\Theta}$ on an object $(B,X)\in \Grothop{F}$ is
		$$\groth{\Theta}_{(B,X)}=\left(\id{B},\Theta_{B,X}\right).$$
		It is straightforward to show that $\groth{-}$ is indeed a $2$-functor.
	\end{proof}

	\begin{rem}\label{remarchetypalthreedimclassifproc}
		\nameit{Theorem}\ref{grothconstrislaxcomma} shows in which sense the $2$-$\catfont{Set}$-enriched Grothendieck construction can be thought of as the archetypal $3$-dimensional classifier, in the sense of a would-be elementary $3$-topos theory. Weber proposed in~\cite{weber_yonfromtwotop} to convert, in the passage from dimension $1$ to dimension $2$, all the monomorphisms into discrete opfibrations. As we said in the introduction, also keeping the classifier to have the terminal as domain and thus be the inclusion of a verum inside of generalized truth values but changing the classification process to be regulated by comma objects is equally good, and actually preferable. Now, in order to reach the dimension $3$, we propose to either upgrade the classification process into one regulated by lax comma objects (as defined here in \nameit{Definition}\ref{completeunivproplaxcomma}) or to consider pullbacks of the notion of fibration that the $2$-$\catfont{Set}$-enriched Grothendieck construction produces. We will describe such notion of fibration in detail below, see \nameit{Definition}\ref{deftwosetopf}.

		It is interesting to notice that we had to move out of $2\h[2]\mbox{-}\CAT$ (see also \nameit{Remark}\ref{needof2catlax}), in order to capture the laxness that permeates the Grothendieck construction (seen already from \nameit{Construction}\ref{2SetGrothconstrtoconicalize}, where we simultaneously constructed the lax normal conical $2$-limits and the $2$-$\catfont{Set}$-enriched Grothendieck construction). So the archetypal elementary $3$-topos seems to be $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$. This idea is also reinforced by the fact that Buckley, in his paper~\cite{buckley_fibredtwocatandbicat}, continuing the work of Bakovi\'c's~\cite{bakovic_fibrofbicat}, found the need of considering trihomomorphisms $F\:\catfont{B}\ensuremath{^{\operatorname{coop}}}\to 2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ in order to capture non-split Hermida's $2$-fibrations (presented in Hermida's~\cite{hermida_somepropoffibasafibredtwocat}) via a suitable Grothendieck construction. So what seems to work is the sequence
		$$\catfont{Set}\quad\leadsto\quad \catfont{CAT} \quad\leadsto\quad 2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$$

		We believe that such sequence is best explained by what we call a \predfn{$2$-$\catfont{V}$-enrichment}, given $\catfont{V}$ a nice enough monoidal category, so that we view the sequence as
		$$\catfont{V}\quad\leadsto\quad \VCAT{\catfont{V}} \quad\leadsto\quad \twoVCAT{\catfont{V}}$$
		with $\catfont{V}=\catfont{Set}$.	Our idea is that enriching again over $\VCAT{\catfont{V}}$ should take into account the fact that $\VCAT{\catfont{V}}$ is a $2$-category, and then be a \predfn{weak enrichment} rather than an ordinary enrichment (that would only consider the underlying category of $\VCAT{\catfont{V}}$). We would like to thank Francesco Dagnino for the interesting discussions that brought to this idea of $2$-$\catfont{V}$-enrichment.

		As described with more detail below, weakly enriching over $\VCAT{\catfont{Set}}=\catfont{CAT}$ actually produces bicategories, lax functors, lax natural transformations and modifications, but we will restrict to consider $2$-categories and $2$-functors, for simplicity. Nevertheless, we will be able to capture the lax natural transformations in this way, and the $2$-$\catfont{V}$-enrichment idea will guide us to propose a notion of colimit and of pointwise Kan extension in the still poorly studied context of $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$, in Section~\ref{sectionkanextensions}. We also notice that the further step in the sequence above could bring towards a version of the Gray-categories, but we have not investigated this yet.
	\end{rem}

	We recall the following remark on the ordinary enrichment.

	\begin{rec}\label{remenrichment}
		If $(\catfont{V},\otimes,I)$ is a monoidal category with coproducts such that $-\otimes -$ preserves coproducts in each variable, we can define a $\catfont{V}$-enriched category as pair $(S,\catfont{A})$ with $S$ a set, that will be the set of objects, and $\catfont{A}$ a monoid in the monoidal category $\m{S\times S}{\catfont{V}}$ of functors (actually given by mere functions) from $S\times S$ to $\catfont{V}$ and natural transformations (actually given by the mere components), that we think of as the monoidal category of square matrices indexed by $S$ with entries in $\catfont{V}$, with the tensor product given by matrix multiplication and tensor unit given by the identity matrix (with $I$ all over the main diagonal and the initial object elsewhere). The multiplication of the monoid $\catfont{A}$ gives indeed the composition of the enriched category, the unit gives the identities and the axioms of monoid precisely ask the composition to be associative and unital.

		We can then also define $\catfont{V}$-enriched functors on this line, using the following construction. Given a $\catfont{V}$-enriched category $(T,\catfont{B})$ and a function $F\:S\to T$, we can define a monoid $F^{\ast} \catfont{B}$ in $\m{S\times S}{\catfont{V}}$, taking
		$$F^{\ast} \catfont{B} =\left(S\times S\ar{F\times F} T\times T \ar{\catfont{B}}\catfont{V}\right)$$
		and defining the multiplication and the unit by whiskering those of $\catfont{B}$ with $F\times F$ on the left. Indeed $F^{\ast}\catfont{B}$ is a monoid, by pasting calculations, since all the required axioms just involve cells of strictly higher levels than $F\times F$.

		Given $(S,\catfont{A})$ and $(T,\catfont{B})$ two $\catfont{V}$-enriched categories, a $\catfont{V}$-enriched functor $(S,\catfont{A})\to (T,\catfont{B})$ can be defined as a pair $(F,\overline{F})$ with $F\:S\to T$ a function and $\overline{F}\:\catfont{A}\to F^{\ast}\catfont{B}$ a morphism between monoids in $\m{S\times S}{\catfont{V}}$.
	\end{rec}

	\begin{rem}
		We now give a weak $2$-categorical generalization of \nameit{Recall}\ref{remenrichment}. The concept of \predfn{weak enrichment} is explored in Garner and Shulman's~\cite{garnershulman_enrichedcatasafreecocompl}, but in the terms presented below (on the line of \nameit{Recall}\ref{remenrichment}) it does not seem to appear in the literature.

		We will use the concepts of pseudomonoid in a $2$-category and of lax morphism between them. A definition of them can be found in Vasilakopoulou's~\cite{vasilakopoulou_monoidalgrothconstr}.
	\end{rem}

	\begin{cons}[Weak enrichment]\label{consweakenrichment}
		Let $(\catfont{K},\otimes,I,\alpha,\lambda,\rho)$ be a monoidal $2$-category, i.e.\ a $2$-category $\catfont{K}$ that is monoidal in the $1$-dimensional sense but such that the tensor product is a $2$-functor $\catfont{K}\times \catfont{K}\to \catfont{K}$. And assume that $\catfont{K}$ has coproducts and that $-\otimes -$ preserves them in each variable. Then, for every set $S$, the $2$-category $\m{S\times S}{\catfont{K}}$ is $2$-monoidal as well, with tensor product given by matrix multiplication and tensor unit given by the identity matrix. Indeed, the matrix multiplication can be extended to a $2$-functor using the $2$-dimensional property of the (now enriched) coproducts, with the $2$-functoriality given by the fact that everything can be discharged on components and that $-\otimes -\:\catfont{K}\times \catfont{K}\to \catfont{K}$ is a $2$-functor.

		We define a \dfn{$\catfont{K}$-weakly enriched category} as a pair $(S,\catfont{A})$ with $S$ a set, thought as the set of objects, and $\catfont{A}$ a pseudomonoid in the monoidal $2$-category $\m{S\times S}{\catfont{K}}$ of square $S$-indexed matrices with entries in $\catfont{K}$ (whose $1$-cells are the $2$-natural transformations and whose $2$-cells are the modifications). Notice that a strict $2$-monoid in $\m{S\times S}{\catfont{K}}$ is the same thing as a monoid in the monoidal category $\m{S\times S}{{\catfont{K}}_{\h[3]0}}$, and thus precisely an enriched category over ${\catfont{K}}_{\h[3]0}$ with object set $S$.

		Now, notice that if $(T,\catfont{B})$ is a $\catfont{K}$-weakly enriched category and $F\:S\to T$ is a function, then
		$$F^{\ast} \catfont{B} =\left(S\times S\ar{F\times F} T\times T \ar{\catfont{B}}\catfont{V}\right)$$
		is a pseudomonoid in $\m{S\times S}{\catfont{K}}$, defining all the needed structure cells as those of $\catfont{B}$ whiskered with $F\times F$ on the left. That $F^{\ast} \catfont{B}$ is a pseudomonoid holds by pasting calculations, since all the required axioms just involve cells of strictly higher levels than $F\times F$.

		Given two $\catfont{K}$-weakly enriched categories $(S,\catfont{A})$ and $(T,\catfont{B})$, we define a \dfn{$\catfont{K}$-weakly enriched functor $(S,\catfont{A})\to (T,\catfont{B})$} as a pair $(F,\overline{F})$ with $F\:S\to T$ a function and $\overline{F}\:\catfont{A}\to F^{\ast}\catfont{B}$ a lax morphism between lax monoids in $\m{S\times S}{\catfont{K}}$.

		Given now $(F,\overline{F}), (G,\overline{G})\:(S,\catfont{A})\to (T,\catfont{B})$ two $\catfont{K}$-weakly enriched functors, we define a \dfn{$\catfont{K}$-weakly enriched natural transformation $\varphi\:(F,\overline{F})\alax{}(G,\overline{G})$} as a collection of $1$-cells
		$$\varphi_A\:I\to \HomC{\catfont{B}}{F(A)}{G(A)}$$
		in $\catfont{K}$ for every $A\in S$ and $2$-cells
		\begin{cd}[2.2][4]
			\& \catfont{A}(A,B) \otimes I \arrow[r, "G\otimes \varphi_{A}",""'{name=A}] \&[1.9ex] \catfont{B}(GA,GB) \otimes \catfont{B}(FA,GA) \arrow[rd, "\opn{comp}"{inner sep=0.3ex}] \\
			\catfont{A}(A,B) \arrow[rd, "\lambda_{\catfont{A}(A,B)}^{-1}"'{inner sep=0.3ex}] \arrow[ru, "\rho_{\catfont{A}(A,B)}^{-1}"{inner sep=0.3ex}] \&\&\&[-3ex] \catfont{B}(FA,GB) \\
			\& I \otimes \catfont{A}(A,B) \arrow[r, "\varphi_{B}\otimes F"',""{name=B}] \& \catfont{B}(FB,GB) \otimes \catfont{B}(FA,FB) \arrow[ru, "\opn{comp}"'{inner sep=0.3ex}]
			\arrow[Rightarrow,from=A,to=B,shift left=2.85ex,"{\h[1]\varphi_{A,B}}",shorten <=2.3ex,shorten >=2ex]
		\end{cd}
		in $\catfont{K}$ for every pair $(A,B)\in S\times S$ such that, for every $A,B,C\in S$, with notations like $\catfont{A}_{A,B}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=}\HomC{\catfont{A}}{A}{B}$ and $\catfont{B}^{F,G}_{A,B}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=}\HomC{\catfont{B}}{F(A)}{G(B)}$ and omitting the tensor product of objects, the pasting in $\catfont{K}$\v[1]
		\newcommand{\z}[0]{}
		\begin{cd}[3][4.75]
			\&\& \catfont{B}^{FG}_{AA}
			\arrow[rrrdd,bend left=20,equal,"{}"]\arrow[rd,"{\lambda^{-1}}"{inner sep=0.2ex}]
			\arrow[d,equal,shift left=5.15ex,shorten <= 3.8ex,shorten >=-2.3ex]
			\\
			I
			\arrow[rru,bend left=18,"{\varphi_A}"{inner sep=0.2ex}]\arrow[rr,"{\id{G(A)}}"]\arrow[rd,"{\id{A}}"'{inner sep=0.2ex}]\&
			\arrow[d,Rightarrow,shift left=0.5ex,"{\overline{G}\h[1]}"'{pos=0.4},shorten <= 0.35ex,shorten >=0.7ex]\&
			\catfont{B}^{GG}_{AA}
			\arrow[rd,"{\rho^{-1}}"{inner sep=0.2ex}]
			\arrow[d,equal,shift left=2.6ex,shorten <= 1.35ex,shorten >=0.35ex]\&
			I\z \catfont{B}^{FG}_{AA}
			\arrow[rd,"{\id{G(A)}\otimes 1}"{inner sep=0.2ex}]
			\arrow[r,iso,shift left=0.2ex,"{\opn{unit}_\catfont{B}^{-1}}"{inner sep=1ex}]\&\vphantom{.}
			\\
			\& \catfont{A}_{AA}
			\arrow[r,"{\rho^{-1}}"]\arrow[ru,"{G}"{inner sep=0.2ex}]\arrow[rd,"{\lambda^{-1}}"'{inner sep=0.2ex}]\&
			\catfont{A}_{AA}\z I
			\arrow[r,"{G\otimes 1}"]\&
			\catfont{B}^{GG}_{AA}\z I
			\arrow[r,"{1\otimes \varphi_A}"]
			\arrow[d,Rightarrow,shift right=4ex,"{\varphi_{A,A}}"{pos=0.495},shorten <= 0.4ex,shorten >=0.17ex]\&
			\catfont{B}^{GG}_{AA}\z \catfont{B}^{FG}_{AA}
			\arrow[r,"{\opn{comp}}"]\&
			\catfont{B}^{FG}_{AA}
			\\
			\&\& I\z \catfont{A}_{AA}
			\arrow[r,"{1\otimes F}"']\&
			I\z \catfont{B}^{FF}_{AA}
			\arrow[r,"{\varphi_A\otimes 1}"']\&
			\catfont{B}^{FG}_{AA}\z \catfont{B}^{FF}_{AA}\v[2]
			\arrow[ru,"{\opn{comp}}"'{inner sep=0.2ex}]
		\end{cd}
		is equal to the pasting\v[1]
		\begin{cd}[3][4.75]
			\&\& \catfont{B}^{FG}_{AA}
			\arrow[rrrd,equal,bend left=16]\arrow[rd,"{\rho^{-1}}"{inner sep=0.2ex}]
			\arrow[d,equal,shift right=5ex,shorten <= 1ex,shorten >=0.7ex]
			\\
			I
			\arrow[rru,bend left=17,"{\varphi_A}"{inner sep=0.2ex}] \arrow[rrd,"{\id{F(A)}}"'{inner sep=0.1ex}]\arrow[rdd,"{\id{A}}"'{inner sep=0.2ex}]\arrow[rr,"{\rho^{-1}}"]\&
			\arrow[dd,Rightarrow,shift left=0.5ex,"{\overline{F}\h[1]}"'{pos=0.575},shorten <= 4.9ex,shorten >=2.4ex]\&
			I\z I
			\arrow[rd,"{1\otimes \id{F(A)}}"'{pos=0.4,inner sep=0.1ex}]\arrow[r,"{\varphi_A\otimes 1}"]
			\arrow[d,equal,shift right=4ex,shorten <= 0.67ex,shorten >=1.3ex]\&
			\catfont{B}^{FG}_{AA}\z I
			\arrow[r,"{1\otimes \id{F(A)}}"]
			\arrow[r,iso,shift left=3.9ex,"{\opn{unit}_\catfont{B}^{-1}}"{inner sep=1ex},start anchor={[xshift=-3ex]}]
			\arrow[d,equal,shift left=3.5ex,shorten <=-0.2ex,shorten >=1.65ex]\&
			\catfont{B}^{FG}_{AA}\z \catfont{B}^{FF}_{AA}
			\arrow[r,"{\opn{comp}}"']\&
			\catfont{B}^{FG}_{AA}
			\\
			\&\&
			\catfont{B}^{FF}_{AA}
			\arrow[r,"{\lambda^{-1}}"']\&
			I\z \catfont{B}^{FF}_{AA}
			\arrow[ru,"{\varphi_A\otimes 1}"'{inner sep=0.2ex}]
			\\
			\&\catfont{A}_{AA}\v[2]
			\arrow[ru,"{F}"']
		\end{cd}
		and the pasting in $\catfont{K}$\v[1]
		\begin{cd}[3.5][-4.9]
			\&[0.2ex] \catfont{A}_{BC}\z\left(\catfont{A}_{AB}\z I\right)
			\arrow[rr,"{1\otimes G\otimes \varphi_A}"]\&[0.6ex]\&[-0.4ex]
			\catfont{A}_{BC}\z\left(\catfont{B}^{GG}_{AB}\z \catfont{B}^{FG}_{AA}\right)
			\arrow[rr,"{1\otimes\opn{comp}}"]
			\arrow[d,Rightarrow,shift right=7.7ex,"{1\otimes \varphi_{A,B}}"{pos=0.4},shorten <= 0.25ex,shorten >=1.68ex]\&[-0.85ex]\&[-0.65ex]
			\catfont{A}_{BC}\z \catfont{B}^{FG}_{AB}
			\arrow[rr,"{G\otimes 1}"]
			\arrow[d,equal,shift left=6ex,shorten <= 0.87ex,shorten >=2.9ex]\&[-0.65ex]\&[0.35ex]
			\catfont{B}^{GG}_{BC}\z\catfont{B}^{FG}_{AB}
			\arrow[rdd,"{\opn{comp}}"{inner sep=0.2ex}]
			\\
			\catfont{A}_{BC}\z\catfont{A}_{AB}
			\arrow[ru,"{1\otimes \rho^{-1}}"{pos=0.47,inner sep=0.14ex}]\arrow[rr,"{1\otimes \lambda^{-1}}"] \arrow[rrd,"{1\otimes F}"{inner sep=0.2ex}]\arrow[rdd,"{F\otimes F}"{inner sep=0.2ex}]\arrow[ddd,"{\opn{comp}}"'{inner sep=0.2ex}]\&\&
			\catfont{A}_{BC}\z \left(I\z \catfont{A}_{AB}\right)
			\arrow[rr,"{1\otimes \varphi_B\otimes F}"] \&\vphantom{.}\&
			\catfont{A}_{BC}\z\left(\catfont{B}^{FG}_{BB}\z\catfont{B}^{FF}_{AB}\right)
			\arrow[rr,"{G\otimes 1\otimes 1}"]\arrow[ru,"{1\otimes \opn{comp}}"{inner sep=0.2ex}]
			\arrow[d,equal,shift right=8ex,shorten <= 0.9ex,shorten >=0.95ex]\&\vphantom{.}\&
			\catfont{B}^{GG}_{BC}\z\left(\catfont{B}^{FG}_{BB}\z\catfont{B}^{FF}_{AB}\right)
			\arrow[ru,"{1\otimes \opn{comp}}"{inner sep=0.2ex}]
			\\
			\&\& \catfont{A}_{B,C}\z\catfont{B}^{FF}_{AB}
			\arrow[rr,"{\rho^{-1}\otimes 1}"]\arrow[rd,"{\lambda^{-1}\otimes 1}"'{inner sep=0.2ex}]\&\&
			\left(\catfont{A}_{BC}\z I\right) \z \catfont{B}^{FF}_{AB}
			\arrow[rr,"{G\otimes \varphi_B\otimes 1}"]
			\arrow[d,Rightarrow,shift right=0ex,"{\varphi_{B,C}\otimes 1}"{pos=0.43},shorten <= 0.77ex,shorten >=1.33ex]\&\&
			\left(\catfont{B}^{GG}_{BC}\z \catfont{B}^{FG}_{BB}\right) \z \catfont{B}^{FF}_{AB}
			\arrow[u,"{\alpha}"]\arrow[rd,"{\opn{comp}\otimes 1}"'{pos=0.4,inner sep=0.2ex}]
			\arrow[rr,iso,shift left=3.24ex,"{\opn{assoc}_\catfont{B}^{-1}}"{inner sep=1ex},start anchor={[xshift=-4ex]}]\&\&
			\catfont{B}^{FG}_{AC}
			\\
			\& \catfont{B}^{FF}_{BC}\z \catfont{B}^{FF}_{AB}
			\arrow[rd,"{\opn{comp}}"{inner sep=0.2ex}]
			\arrow[d,Rightarrow,shift right=1.6ex,"{\overline{F}\h[1]}"'{pos=0.42},shorten <= 0.81ex,shorten >=1.29ex]
			\arrow[rr,iso,shift right=4.36ex,"{\opn{assoc}_\catfont{B}}"{inner sep=1ex},start anchor={[xshift=7.2ex]}]\&\&
			\left(I \z \catfont{A}_{BC}\right)\z \catfont{B}^{FF}_{AB}
			\arrow[rr,"{\varphi_C\otimes F\otimes 1}"']\&\vphantom{.}\&
			\left(\catfont{B}^{FG}_{CC}\z \catfont{B}^{FF}_{BC}\right)\z \catfont{B}^{FF}_{AB}
			\arrow[rr,"{\opn{comp}\otimes 1}"']\&\&
			\catfont{B}^{FG}_{BC}\z \catfont{B}^{FF}_{AB}
			\arrow[ru,"{\opn{comp}}"{inner sep=0.2ex}]
			\\[-0.1ex]
			\catfont{A}_{AC}
			\arrow[rr,"{F}"']\&\vphantom{.}\&
			\catfont{B}^{FF}_{AC}
			\arrow[rr,"{\lambda^{-1}}"']\&\&
			I\z \catfont{B}^{FF}_{AC}
			\arrow[rr,"{\varphi_C\otimes 1}"']\&\&
			\catfont{B}^{FG}_{CC}\z \catfont{B}^{FF}_{AC}\v[2]\arrow[rruu,bend right=55,"{\opn{comp}}"{pos=0.37,inner sep=0.2ex}]
		\end{cd}
		is equal to the pasting\v[1]
		\begin{cd}[3.5][-0.9]
			\& \catfont{A}_{BC}\z\left(\catfont{A}_{AB}\z I\right)
			\arrow[rr,"{1\otimes G\otimes \varphi_A}"]\&[0.2ex]\&[0.2ex]
			\catfont{A}_{BC}\z\left(\catfont{B}^{GG}_{AB}\z \catfont{B}^{FG}_{AA}\right)
			\arrow[rr,"{1\otimes\opn{comp}}"]\&[-0.2ex]\&[-0.2ex]
			\catfont{A}_{BC}\z \catfont{B}^{FG}_{AB}
			\arrow[rrd,bend left=17,"{G\otimes 1}"{inner sep=0.2ex}]\&[-0.2ex]\&[-0.2ex]
			\\
			\& \catfont{B}^{GG}_{BC}\z\catfont{B}^{GG}_{AB}
			\arrow[rr,"{\opn{comp}}"]
			\arrow[d,Rightarrow,shift left=4.5ex,"{\overline{G}\h[1]}"'{pos=0.37},shorten <= 0.43ex,shorten >=1.42ex]\&\&
			\catfont{B}^{GG}_{AC}
			\arrow[rr,"{\rho^{-1}}"]
			\arrow[d,equal,shift left=9.5ex,shorten <= 0.79ex,shorten >=2.76ex]\&\&
			\catfont{B}^{GG}_{AC}\z I
			\arrow[rd,"{1\otimes \varphi_A}"{inner sep=0.2ex}]
			\arrow[rr,iso,shift left=0.5ex,"{\opn{assoc}_\catfont{B}^{-1}}"{inner sep=1ex},start anchor={[xshift=-0.58ex]}]\&\&
			\catfont{B}^{GG}_{BC}\z\catfont{B}^{FG}_{AB}
			\arrow[dd,"{\opn{comp}}"]
			\\
			\catfont{A}_{BC}\z\catfont{A}_{AB}
			\arrow[ruu,bend left=25,"{1\otimes \rho^{-1}}"{inner sep=0.12ex}] \arrow[ru,"{G\otimes G}"{inner sep=0.2ex}]
			\arrow[rr,"{\opn{comp}}"']\&\vphantom{.}\&
			\catfont{A}_{AC}
			\arrow[rr,"{\rho^{-1}}"]\arrow[ru,"{G}"{inner sep=0.2ex}]\arrow[rd,"{\lambda^{-1}}"'{inner sep=0.2ex}]\&\vphantom{.}\&
			\catfont{A}_{AC}\z I
			\arrow[rr,"{G\otimes \varphi_A}"]
			\arrow[d,Rightarrow,shift left=0.5ex,"{\varphi_{A,C}}"{pos=0.43},shorten <= 0.75ex,shorten >=1.235ex]\&\&
			\catfont{B}^{GG}_{AC}\z \catfont{B}^{FG}_{AA}
			\arrow[rd,"{\opn{comp}}"{inner sep=0.2ex}]
			\\
			\&\&\& I\z \catfont{A}_{AC}
			\arrow[rr,"{\varphi_C\otimes F}"']\&\vphantom{.}\&
			\catfont{B}^{FG}_{CC}\z \catfont{B}^{FF}_{AC}
			\arrow[rr,"{\opn{comp}}"']\&\&
			\catfont{B}^{FG}_{AC}\v[1.5]
		\end{cd}
	\end{cons}

	\begin{rem}
		We observe that the definition of weakly enriched natural transformation we have given in \nameit{Construction}\ref{consweakenrichment} cannot be easily generalized to a fully lax enriched context in which we consider lax monoids rather than pseudomonoids in $\m{S\times S}{\catfont{K}}$. Indeed the pastings we need to consider have to use the associator of $\catfont{B}$ in both the directions.

		We also notice that \nameit{Construction}\ref{consweakenrichment} is particularly useful when $\catfont{K}$ is $\VCAT{\catfont{V}}$, since we then obtain a construction that we can apply to a starting ordinary $\catfont{V}$ (creating further dimension levels that did not initially belong to $\catfont{V}$).
	\end{rem}

	\begin{defne}[The $2$-$\catfont{V}$-enrichment]\label{deftwovenrichment}
		Let $\catfont{V}$ be a nice enough monoidal category such that $\VCAT{\catfont{V}}$ with the tensor product of $\catfont{V}$-categories (so we need at least a braiding in $\catfont{V}$ to have such tensor product) becomes a monoidal $2$-category with coproducts such that its tensor product preserves them in each variable.

		We call \dfn{$2$-$\catfont{V}$-enriched category (resp. functor, natural transformation)} a $\left(\VCAT{\catfont{V}}\right)$-weakly enriched category (resp. functor, natural transformation). We call $\twoVCAT{\catfont{V}}$ the tridimensional structure they form.
	\end{defne}

	\begin{exampl}\label{exampletwosetenrichment}
		Consider $\catfont{V}=\catfont{Set}$. Then $2$-$\catfont{Set}$-enriched categories, functors and natural transformations are, respectively, bicategories, lax functors and lax natural transformations. A notion of $2$-$\catfont{V}$-enriched modification could be given as well, such that the $2$-$\catfont{Set}$-enriched modifications are precisely the modifications between lax natural transformations.

		So we can now think of $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ as the lax $3$-category of $2$-$\catfont{Set}$-enriched categories, restricted to strict weakly enriched categories and functors for simplicity.
	\end{exampl}

	\begin{rem}\label{remnametwosetenrichedgroth}
		We can now clarify the name ``$2$-$\catfont{Set}$-enriched Grothendieck construction", after \nameit{Example}\ref{exampletwosetenrichment}. We have given this name from the observation that it is the Grothendieck construction that lives in the $2$-$\catfont{Set}$-enriched world (it could also be extended to consider bicategories and lax functors into $\catfont{CAT}$). It takes enriched presheaves with values in $\VCAT{\catfont{Set}}=\catfont{CAT}$, that is the base of the weak enrichment we consider, and converts them into an extended notion of Grothendieck fibration, that is worth to be called \predfn{$2$-$\catfont{Set}$-fibration}.

		We think that the $2$-$\catfont{Set}$-enriched Grothendieck construction should be taken into consideration towards the development of a general enriched Grothendieck construction. This will be explored in future work. In particular, we believe that for this the $2$-$\catfont{Set}$-enrichment should be distinguished from the ordinary $\catfont{Cat}$-enrichement, and we expect the latter to be much less useful than the former.
	\end{rem}

	\begin{rem}
		We now proceed to describe the \predfn{$2$-$\catfont{Set}$-opfibrations} over a $2$-category $\catfont{B}$, that are the notion of opfibration produced by the $2$-$\catfont{Set}$-enriched Grothendieck construction, in the sense that they will precisely be (up to restricting to small fibres) what forms the essential image of the $2$-functor $\groth{-}\:\m{\catfont{B}}{\catfont{CAT}}\to \slice{2\h[2]\mbox{-}\CAT}{\catfont{B}}$ described in \nameit{Proposition}\ref{twosetgrothcanonicallyextendstwofunctor}. They have been firstly introduced by Lambert in~\cite{lambert_discretetwofib} with the name ``discrete $2$-fibrations" (up to co-op changes), justified by the fact that they are equivalently the locally discrete Hermida's $2$-fibrations (for Hermida's $2$-fibrations, see~\cite{hermida_somepropoffibasafibredtwocat}). We believe, however, that the \predfn{$2$-$\catfont{Set}$-fibrations} are what fully realizes the concept of fibration in dimension $2$, while Hermida's $2$-fibrations belong more to dimension $3$.

		We shall look at \nameit{Construction}\ref{2SetGrothconstrtoconicalize} (where we intuitively produced the $2$-$\catfont{Set}$-enriched Grothendieck construction) to investigate which lifting properties hold for $\groth{F}$ for some $2$-functor $F\:\catfont{B}\to \catfont{CAT}$ and could characterize the functors that are isomorphic to some $\groth{F}$. Since, calling $H_0$ the underlying functor of a $2$-functor $H$, we have that
		$$\groth{F}_0=\groth{F_0},$$
		we obtain that $\groth{F}_0$ is a usual Grothendieck opfibration. In the notation of \nameit{Construction}\ref{2SetGrothconstrtoconicalize}, the cartesian liftings are given by the $\underline{f}^X$. But we now also have liftings of the $2$-cells in $\catfont{B}$. Indeed, for every $2$-cell $\delta\:f\aR{}g\:B\to B'$ in $\catfont{B}$ and every $X\in F(B)$, we have a $2$-cell in $\Grothop{F}$
		$$\underline{\delta}^X\:(f,F(\delta)_X)\aR{}(g,\id{})\:(B,X)\to (B',F(g)(X))$$
		such that $\groth{F}(\underline{\delta}^X)=\delta$. As a consequence, for every morphism $(g,\beta)$ in $\Grothop{F}$, the $2$-cell
		$$\delta=(\id{B},\beta)\h \underline{\delta}^X$$
		in $\Grothop{F}$ is a $2$-cell that lifts $\delta$ to the morphism $(g,\beta)$ that lives above the codomain $g$ of $\delta$. Recall from \nameit{Construction}\ref{2SetGrothconstrtoconicalize} that all the $2$-cells in $\Grothop{F}$ emerge in this way. As we noticed there, having a $2$-cell $\delta$ in $\Grothop{F}$ as above is actually a property for $\delta$. Thus, fixed $\delta\:f\aR{}g$ in $\catfont{B}$ and $(g,\beta)$ in $\Grothop{F}$ that lives above the codomain $g$ of $\delta$, the liftings of $\delta$ to $(g,\beta)$ are unique. And we conclude that $\groth{F}$, further than being an ordinary Grothendieck opfibration, is also locally a discrete fibration (see \nameit{Definition}\ref{deftwosetopf}).

		It is interesting to notice the change of direction from opfibration on objects to fibration locally. This is given by the fact that we are classifying from $\catfont{1}\:\catfont{1}\to \catfont{CAT}$ using lax comma objects. We will see in \nameit{Remark}\ref{coopflavourstwosetenrichedgroth} that we have other classifiers as well, with all the co-op flavours, but we believe that this is the most natural one for what we want to call \predfn{$2$-$\catfont{Set}$-opfibration}.
	\end{rem}

	\begin{defne}\label{deftwosetopf}
		Let $\catfont{B}$ be a $2$-category. A \dfn{$2$-$\catfont{Set}$-opfibration over $\catfont{B}$} (or \dfn{discrete $2$-fibration} in the language of Lambert's~\cite{lambert_discretetwofib}) is a $2$-functor $P\:\catfont{E}\to \catfont{B}$ such that
		\begin{enum}
			\item the underlying functor $P_0$ of $P$ is an ordinary Grothendieck opfibration;
			\item for every pair $X,Y\in \catfont{E}$ the functor
			$$P_{X,Y}\:\HomC{\catfont{E}}{X}{Y}\to \HomC{\catfont{B}}{P(X)}{P(Y)}$$
			is a discrete fibration.
		\end{enum}
		We say that $P$ is \dfn{split} if $P_0$ is so.
	\end{defne}

	\begin{rem}
		The following theorem is proved in Lambert's~\cite{lambert_discretetwofib}. One can find there some examples of $2$-$\catfont{Set}$-opfibrations as well.

		We will extend such theorem to a complete $2$-equivalence betwen $\m{\catfont{B}}{\catfont{CAT}}$ with various laxness flavours on morphisms and corresponding $2$-categories of $2$-$\catfont{Set}$-opfibrations in Section~\ref{sectionkanextensions}, after showing that the lax comma object square of \nameit{Theorem}\ref{grothconstrislaxcomma} exhibits a \predfn{weak Kan extension} in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$.
	\end{rem}

	\begin{teor}\label{essentialimagegivenbytwosetopf}
		Let $\catfont{B}$ be a $2$-category. The essential image of the $2$-functor
		$$\groth{-}\:\m{\catfont{B}}{\catfont{CAT}}\to \slice{2\h[2]\mbox{-}\CAT}{\catfont{B}}$$
		is given by the split $2$-$\catfont{Set}$-opfibrations with small fibres.
	\end{teor}

	\begin{rem}\label{coopflavourstwosetenrichedgroth}
		The following table shows the four co-op versions of the $2$-$\catfont{Set}$-enriched Grothendieck construction, with the corresponding notions of fibration. We have named the various notions of fibration from an enriched point of view, where the op always makes sense while the co does not.
		\begin{center}\h[-18]
			\begin{minipage}{3.3cm}
				\begin{eqD*}
					\h[-7]\sq*[l][5.5][4.5][\lax \opn{comma}][2.7][2.2][0.56]{\Grothop{F}}{\catfont{1}}{\catfont{B}}{\catfont{CAT}}{}{\groth{F}}{\catfont{1}}{F}
				\end{eqD*}
				\begin{center}{\footnotesize $2$-$\catfont{Set}$-opfibrations:}

					\noindent {\footnotesize opfibrations, locally discrete fibrations}
				\end{center}
			\end{minipage}\quad
			\begin{minipage}{3.38cm}
				\begin{eqD*}
					\h[-7]\sq*[o][5.5][4.5][\lax \opn{comma}][2.2][2.7][0.6]{\Groth{F}}{\catfont{1}}{\catfont{B}}{\catfont{CAT}\ensuremath{^{\operatorname{op}}}}{}{\groth{F}}{\catfont{1}}{F}
				\end{eqD*}
				\begin{center}{\footnotesize $2$-$\catfont{Set}$-fibrations:}

					\noindent {\footnotesize fibrations, locally discrete opfibrations}
				\end{center}
			\end{minipage}\quad
			\begin{minipage}{3.47cm}
				\begin{eqD*}
					\h[-7]\sq*[l][5.5][3.4][\oplax \opn{comma}][2.7][2.2][0.56]{\Grothcoop{F}}{\catfont{1}}{\catfont{B}}{\catfont{CAT}\ensuremath{^{\operatorname{co}}}}{}{\groth{F}}{\catfont{1}}{F}
				\end{eqD*}
				\begin{center}{\footnotesize $2$-$\catfont{Set}$-coopfibrations:}

					\noindent {\footnotesize opfibrations, locally discrete opfibrations}
				\end{center}
			\end{minipage}\quad
			\begin{minipage}{3.53cm}
				\begin{eqD*}
					\h[-7]\sq*[o][5.5][3.3][\oplax \opn{comma}][2.2][2.7][0.6]{\Grothco{F}}{\catfont{1}}{\catfont{B}}{\catfont{CAT}\ensuremath{^{\operatorname{coop}}}}{}{\groth{F}}{\catfont{1}}{F}
				\end{eqD*}
				\begin{center}{\footnotesize $2$-$\catfont{Set}$-cofibrations:}

					\noindent {\footnotesize fibrations, locally discrete fibrations}
				\end{center}
			\end{minipage}
		\end{center}
	\end{rem}

	\section{A pointwise Kan extension result}\label{sectionkanextensions}

	In this section, we investigate a pointwise Kan extension result for the $2$-$\catfont{Set}$-enriched Grothendieck construction. The difficulty we encounter is that no notion of (pointwise) Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ seems to appear in the literature. The pointwise version in particular is hard to establish, as it requires a concept of colimit ``internal to $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$", i.e.\ in a $2$-$\catfont{Set}$-category, that does not seem to have been introduced yet.

	We firstly give a definition of weak Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ (actually, in any lax $3$-category), and prove the weak version of the pointwise Kan extension result we aim at establishing. This weak result will be enough to prove $2$-fully faithfulness results for the $2$-$\catfont{Set}$-enriched Grothendieck construction (in three laxness flavours) and complete \nameit{Theorem}\ref{essentialimagegivenbytwosetopf} to $2$-equivalences between $2$-copresheaves (on a $2$-category) and $2$-categories of $2$-$\catfont{Set}$-opfibrations.

	Then, we propose a notion of ``internal colimit in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$", i.e.\ of colimit in a $2$-$\catfont{Set}$-enriched category, and originally define the concept of \predfn{pointwise left Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ along a $2$-$\catfont{Set}$-opfibration} using it. We prove that the lax comma object square produced by the $2$-$\catfont{Set}$-enriched Grothendieck construction (\nameit{Theorem}\ref{grothconstrislaxcomma}) exhibits a \predfn{pointwise left Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$}. This is the third main result of this paper, presented in \nameit{Theorem}\ref{teorpointkanextforgroth}.

	In the last part of this section, we show the fourth main result of this paper, which is that a pointwise left Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ (as defined here) is always a weak one as well. This (presented in \nameit{Proposition}\ref{pointkanisalsoweak}) will be based on an $\oplaxn\mbox{-}\lax$ generalization of the parametrized Yoneda lemma that does not seem to appear in the literature. Such lemma, proved in \nameit{Theorem}\ref{parametrizedyonedaoplaxnlax}, also sheds more light on the concept of lax normal natural transformation (\nameit{Definition}\ref{deflaxnormal}). Indeed it shows how the oplax normal naturality is the minimum amount of strictness that is required to expand the data on the identities to a lax natural transformation.

	\begin{defne}\label{defweakkanextensionlaxthreecat}
		A diagram
		\begin{cd}[7][5]
			\catfont{B} \arrow[rr,"{F}"]\arrow[d,"{K}"']\&\arrow[dl,Rightarrow,shift right=0.75ex,"{\lambda}"{pos=0.355, inner sep= 0.35ex},shorten <=1.65ex,shorten >= 3.35ex]\& \catfont{C} \\
			\catfont{A}\arrow[rru,"L"']
		\end{cd}
		in a lax $3$-category $\catfont{Q}$ (that is a category enriched over the $1$-category of $2$-categories and lax functors), exhibits $L$ as the \dfn{weak left Kan extension of $F$ along $K$}, written $L={\lan{K}{F}}$, if pasting with $\lambda$ gives an isomorphism of categories
		\begin{equation}\label{eqweakkanextensionlaxthreecat}
			\HomC{\HomC{\catfont{Q}}{\catfont{A}}{\catfont{C}}}{L}{U}\cong \HomC{\HomC{\catfont{Q}}{\catfont{B}}{\catfont{C}}}{F}{U\circ K}
		\end{equation}
		for every $U\in \HomC{\catfont{Q}}{\catfont{A}}{\catfont{C}}$ (the $2$-naturality in $U$ is granted automatically).
	\end{defne}

	\begin{rem}\label{remhomtwocategoriestwocatlax}
		Lambert showed in~\cite{lambert_discretetwofib} that $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ is a lax $3$-category with hom-$2$-categories $$\HomC{2\h[2]\mbox{-}\CAT_{\h[-3]\lax}}{\catfont{A}}{\catfont{C}}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \mlax{\catfont{A}}{\catfont{C}}$$
		where $\mlax{\catfont{A}}{\catfont{C}}$ is the $2$-category of $2$-functors from $\catfont{A}$ to $\catfont{C}$, lax natural transformations and modifications.

		So, in the notation of \nameit{Definition}\ref{defweakkanextensionlaxthreecat}, the isomorphism of categories of equation~\refs{eqweakkanextensionlaxthreecat} becomes, for every $U\in \mlax{\catfont{A}}{\catfont{C}}$,
		$$\HomC{\mlax{\catfont{A}}{\catfont{C}}}{L}{U}\cong \HomC{\mlax{\catfont{B}}{\catfont{C}}}{F}{U\circ K}.$$

	\end{rem}

	\begin{prop}\label{isomorphismsweakkanextension}
		Let $F\:\catfont{A}\to \catfont{CAT}$ be a $2$-functor and consider its $2$-$\catfont{Set}$-enriched Grothendieck construction
		\sq[l][7][7][\lax \opn{comma}][2.7][2.2][0.65]{\Grothop{F}}{\catfont{1}}{\catfont{A}}{\catfont{CAT}}{}{\groth{F}}{\catfont{1}}{F}
		Call $\lambda^F$ the lax normal natural transformation that presents such lax comma object in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$. Then, for every $U\:\catfont{A}\to \catfont{CAT}$, pasting with $\lambda^F$ gives an isomorphism of categories
		$$\HomC{\mlax{\catfont{A}}{\catfont{CAT}}}{F}{U}\cong \HomC{\mlax{\Grothopdiag{F}}{\catfont{CAT}}}{\Delta 1}{U\circ \groth{F}},$$
		where $\Delta 1\:\Grothop{F}\to \catfont{CAT}$ is the $2$-functor constant at $1$. Moreover, this restricts to isomorphisms
		$$\HomC{\mps{\catfont{A}}{\catfont{CAT}}}{F}{U}\cong \HomC{\msigma{\Grothopdiag{F}}{\catfont{CAT}}}{\Delta 1}{U\circ \groth{F}}$$
		$$\HomC{\m{\catfont{A}}{\catfont{CAT}}}{F}{U}\cong \HomC{\mlaxn{\Grothopdiag{F}}{\catfont{CAT}}}{\Delta 1}{U\circ \groth{F}},$$
		where $\opn{ps}$ means to restrict to pseudonatural transformations and $\opn{sigma}$ means to restrict to \predfn{sigma natural transformations}, that have been defined in Descotte, Dubuc and Szyld's paper~\cite{descottedubucszyld_sigmalimandflatpseudofun} and are a pseudo version of the lax normal natural transformations \pteor{asking the special structure $2$-cells to be isomorphisms rather than identities}.
	\end{prop}
	\begin{proof}
		The first isomorphism is proved in Bird's PhD thesis~\cite{bird_limitsintwocatoflocpresented}. Looking at the explicit construction, it is straightforward to see that it restricts to the other two. In this, what works very well is that $\lambda^F$ is a lax normal natural transformation, and thus always contributes with an identity in the special structure $2$-cells.
	\end{proof}

	\begin{rem}
		The third isomorphism of \nameit{Proposition}\ref{redlaxnormalconical} offers a shorter but less intuitive and elementary proof to \nameit{Theorem}\ref{redlaxnormalconical} (reduction of the weighted $2$-limits to lax normal conical ones). Indeed this is what Street did in~\cite{street_limitsindexedbycatvalued}.
	\end{rem}

	\begin{teor}\label{corollweakkanextension}
		Let $F\:\catfont{B}\to \catfont{CAT}$ be a $2$-functor. Then the lax normal natural transformation $\lambda^F$ of \nameit{Proposition}\ref{isomorphismsweakkanextension} exhibits
		$$F=\lan{\groth{F}}{\Delta 1}.$$
	\end{teor}
	\begin{proof}
		The proof is a combination of \nameit{Remark}\ref{remhomtwocategoriestwocatlax} and \nameit{Proposition}\ref{isomorphismsweakkanextension}.
	\end{proof}

	\begin{rem}
		Exactly as can be done in dimension $1$ (see \nameit{Remark}\ref{remtheorkanextisuseful}), we can use \nameit{Theorem}\ref{corollweakkanextension} (or actually \nameit{Proposition}\ref{isomorphismsweakkanextension} to prove the $2$-fully faithfulness of the $2$-$\catfont{Set}$-enriched Grothendieck construction (in three laxness flavours) and complete \nameit{Theorem}\ref{essentialimagegivenbytwosetopf} to $2$-equivalences between $2$-copresheaves and $2$-$\catfont{Set}$-opfibrations. The fact that the first $2$-functor $\groth{-}$ of \nameit{Theorem}\ref{twofullyfaithfulnessgroth} is $2$-fully faithful is proved also in Bird's~\cite{bird_limitsintwocatoflocpresented}, but without mentioning that it comes from a weak Kan extension result. None of the three $2$-equivalence results of \nameit{Theorem}\ref{twofullyfaithfulnessgroth} seems to appear in the literature.
	\end{rem}

	\begin{teor}\label{twofullyfaithfulnessgroth}
		Let $\catfont{A}$ be a $2$-category. The $2$-$\catfont{Set}$-enriched Grothendieck construction \pteor{extended to consider lax natural transformations as in the proof of \nameit{Proposition}\ref{twosetgrothcanonicallyextendstwofunctor}} produces a $2$-equivalence
		$$\groth{-}\:\mlax{\catfont{A}}{\catfont{CAT}}\ensuremath{\stackrel{\raisebox{-1ex}{\kern-.3ex$\scriptstyle\sim$}}{\rightarrow}} \Fib[t][n][\catfont{A}]$$
		where $\Fib[t][n][\catfont{A}]$ is the full sub-$2$-category of $\h[1]\slice{2\h[2]\mbox{-}\CAT}{\catfont{A}}$ given by the split $2$-$\catfont{Set}$-opfibrations with small fibres. Moreover this restricts to $2$-equivalences
		$$\groth{-}\:\mps{\catfont{A}}{\catfont{CAT}}\ensuremath{\stackrel{\raisebox{-1ex}{\kern-.3ex$\scriptstyle\sim$}}{\rightarrow}} \Fib'[t][n][\catfont{A}]$$
		$$\groth{-}\:\m{\catfont{A}}{\catfont{CAT}}\ensuremath{\stackrel{\raisebox{-1ex}{\kern-.3ex$\scriptstyle\sim$}}{\rightarrow}} \Fib"[t][n][\catfont{A}]$$
		where $\Fib'[t][n][\catfont{A}]$ and $\Fib"[t][n][\catfont{A}]$ restrict $\Fib[t][n][\catfont{A}]$ respectively\v[0.4] to cartesian functors \pteor{as underlying functors of the $1$-cells} and to cleavage preserving functors.
	\end{teor}
	\begin{proof}
		We already know by \nameit{Theorem}\ref{essentialimagegivenbytwosetopf} that the essential image of $\groth{-}$ (in each of the three versions) is given by the split $2$-$\catfont{Set}$-opfibrations. So we are missing the $2$-fully faithfulness of the three $2$-functors.

		Let $F,G\:\catfont{A}\to \catfont{CAT}$ be $2$-functors. Then combining \nameit{Proposition}\ref{isomorphismsweakkanextension} (or the weak Kan extension result \nameit{Theorem}\ref{corollweakkanextension}) and \nameit{Theorem}\ref{grothconstrislaxcomma} (the $2$-$\catfont{Set}$-enriched Grothendieck construction is a lax comma object in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$) we obtain the composite isomorphism of categories
		$$\HomC{\mlax{\catfont{A}}{\catfont{CAT}}}{F}{G}\cong \HomC{\mlax{\Grothopdiag{F}}{\catfont{CAT}}}{\Delta 1}{G\circ \groth{F}}\cong \HomC{\slice{2\h[2]\mbox{-}\CAT}{\catfont{A}}}{\Grothopdiag{F}}{\Grothopdiag{G}}$$
		where the first functor is given by pasting with the $2$-cell $\lambda^F$ that presents the lax comma object $\Grothop{F}$ and the second functor is the inverse of pasting with $\lambda^G$ on objects and producing the modification associated to the lax interchange rule (see \nameit{Remark}\ref{needof2catlax}) on morphisms. Indeed the second functor is surely a bijection on objects, and it is also a bijection on morphisms by part $(ii)$ of \nameit{Definition}\ref{completeunivproplaxcomma} (the $2$-dimensional part of the universal property of the lax comma obejct) with $\Gamma=\id{}$ and $\Delta=\id{}$ (and so in particular strict $2$-natural transformations). Since the composite functor precisely coincides with the functor on morphisms associated to $\groth{-}$ between $\catfont{A}$ and $\catfont{C}$ (see the proof of \nameit{Proposition}\ref{twosetgrothcanonicallyextendstwofunctor}), this completes the proof of the first $2$-equivalence.

		The composite isomorphism above then restricts to the following two:
		$$\scalebox{0.95}{$\HomC{\mps{\catfont{A}}{\catfont{CAT}}}{F}{G}\cong \HomC{\msigma{\Grothopdiag{F}}{\catfont{CAT}}}{\Delta 1}{G\circ \groth{F}}\cong \HomC{\Fib'[t][n][\catfont{A}]}{\Grothopdiag{F}}{\Grothopdiag{G}}$}$$
		$$\scalebox{0.975}{$\HomC{\m{\catfont{A}}{\catfont{CAT}}}{F}{G}\cong \HomC{\mlaxn{\Grothopdiag{F}}{\catfont{CAT}}}{\Delta 1}{G\circ \groth{F}}\cong \HomC{\Fib"[t][n][\catfont{A}]}{\Grothopdiag{F}}{\Grothopdiag{G}}$}$$
		by part $(i)$ of \nameit{Definition}\ref{completeunivproplaxcomma}, since whiskering $\lambda^G$ on the left with a $2$-functor $\Grothop{F}\to \Grothop{G}$ looks at the second component of the morphisms in $\Grothop{G}$. Indeed, if we start for example from $\gamma\:\Delta 1 \alaxn{} G\circ \groth{F}$, the associated functor $V^\gamma\:\Grothop{F}\to \Grothop{G}$ is such that, for every morphism $(f,\id{})$ in $\Grothop{F}$, $$\pr{1}(V^\gamma (f,\id{}))=f \quad\text{and}\quad \pr{2}(V^\gamma (f,\id{}))=(\lambda^G\h V^\gamma)_{(f,\id{})}=\gamma_{(f,\id{})}=\id{}.$$
		And then the first $2$-equivalence restricts to the other two.
	\end{proof}

	\begin{rem}
		It is interesting to notice the bond between cartesian or cleavage preserving functors and sigma or lax normal natural transformations. \nameit{Theorem}\ref{twofullyfaithfulnessgroth} also suggests how to define the various flavours of $1$-cells between $2$-$\catfont{Set}$-opfibrations, showing their preasheaf counterparts as well. In particular there are no requests on the morphism part of such $1$-cells, but this perfectly fits the fact that the $2$-$\catfont{Set}$-opfibrations are discrete fibrations on morphisms.
	\end{rem}

	\begin{rem}\label{remaimatpointwise}
		We now aim at proving that, for some notion of \predfn{pointwise Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$}, the weak left Kan extension of \nameit{Theorem}\ref{corollweakkanextension} is \predfn{pointwise} as well. This would mean to have a complete $2$-categorical analogue of \nameit{Theorem}\ref{copresheavesarekanextensions}.

		Firstly, in order to define a \predfn{pointwise Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$}, we need to understand better the notion of (co)limit in the setting of $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$. There surely is a natural notion of ``external limit" in this context, i.e.\ a limit in the whole $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$, which is that of limit enriched over the $1$-category of $2$-categories and lax functors, since $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ is enriched over such $1$-category. Indeed, for $\catfont{C}$ a $2$-category, this recovers the natural isomorphism
		$$\HomC{2\h[2]\mbox{-}\CAT_{\h[-3]\lax}}{\catfont{U}}{\laxcomma{\Id{\catfont{C}}}{\Id{\catfont{C}}}}\cong \HomC{2\h[2]\mbox{-}\CAT_{\h[-3]\lax}}{\catfont{2}}{\HomC{2\h[2]\mbox{-}\CAT_{\h[-3]\lax}}{\catfont{U}}{\catfont{C}}}$$
		presented by Lambert in~\cite{lambert_discretetwofib} as the universal property of the lax comma object $\laxcomma{\Id{\catfont{C}}}{\Id{\catfont{C}}}$ being the power of $\catfont{C}$ by $\catfont{2}$ in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$.

		But we are more interested in an ``internal" notion of colimit, i.e.\ a notion of colimit in a $2$-$\catfont{Set}$-enriched category (after \nameit{Example}\ref{exampletwosetenrichment}). This should include the notion of colimit in a $2$-category, but be able to express the laxness as well. And there is a notion of colimit that fulfils both these requests and naturally emerges in the world of $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ together with the $2$-$\catfont{Set}$-enriched Grothendieck construction: that is, the oplax normal conical $2$-colimit (see \nameit{Remark}\ref{remoplaxnormalvslaxnormal}, \nameit{Definition}\ref{defoplaxnormalconical} and \nameit{Theorem}\ref{redoplaxnormalconical}). Indeed the $2$-$\catfont{Set}$-enriched Grothendieck construction for $2$-functors $\catfont{A}\ensuremath{^{\operatorname{op}}}\to \catfont{CAT}$ still lives in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$, as showed in \nameit{Remark}\ref{coopflavourstwosetenrichedgroth}. But here we will need a slightly more general notion. We propose to use now non-necessarily conical oplax normal $2$-colimits as colimits in a $2$-$\catfont{Set}$-category (see \nameit{Definition}\ref{defoplaxnormaltwocolimit}).

		Notice that there is a structure of marking included in the definition of such colimit, in the sense that further than specifying the diagram and the weight we also need to specify the way in which we view the domain of the diagram as a Grothendieck construction. And actually, we will need to consider both the Grothendieck constructions that live in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$, for $2$-functors of both the forms $\catfont{A}\to\catfont{CAT}$ and $\catfont{A}\ensuremath{^{\operatorname{op}}}\to \catfont{CAT}$ (this will bring to the definition of \predfn{opmarked oplax normal $2$-colimit}).

		Recall that in Section~\ref{sectionlaxnormallimits} we showed that at least the oplax normal conical $2$-colimits are as expressive as the weighted $2$-colimits. But there is a change of perspective here. We prefer specifying a diagram, a weight and a marking rather than condensing the three to some modified diagram and weight.
	\end{rem}

	\begin{defne}\label{defoplaxnormaltwocolimit}
		Let $M\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to \catfont{CAT}$ (the marking), $F\:\Groth{M}\to \catfont{C}$ (the diagram) and $W\:{\left(\Groth{M}\right)}\ensuremath{^{\operatorname{op}}}\to \catfont{CAT}$ (the weight) be $2$-functors with $\catfont{A}$ small.

		The \dfn{oplax normal $2$-colimit of $F$ marked by $M$ and weighted by $W$}, denoted as $\oplaxnmcolim{M}{W}{F}$, is (if it exists) an object $C\in \catfont{C}$ together with an isomorphism of categories
		$$\HomC{\catfont{C}}{C}{U}\cong \HomC{\moplaxn{{\left(\Grothdiag{M}\right)}\ensuremath{^{\operatorname{op}}}}{\catfont{CAT}}}{W}{\HomC{\catfont{C}}{F(-)}{U}}$$
		$2$-natural in $U\in \catfont{C}$.

		\noindent When $\oplaxnmcolim{M}{W}{F}$ exists, taking $U=C$ and considering the identity on $C$ gives us in particular an oplax normal natural transformation
		$$\mu\:W \aoplaxn{}\HomC{\catfont{C}}{F(-)}{C},$$
		called the \dfn{universal oplax normal cocylinder}.

		We will need to consider also the case in which the domain of $F$ is expressed as $\Grothop{M}$ for some $2$-functor $M\:\catfont{A}\to \catfont{CAT}$, and $W\:{\left(\Grothop{M}\right)}\ensuremath{^{\operatorname{op}}}\to \catfont{CAT}$. The \dfn{oplax normal $2$-colimit of $F$ opmarked by $M$ and weighted by $W$}, denoted as $\oplaxnopmcolim{M}{W}{F}$, is (if it exists) an object $C\in \catfont{C}$ together with an isomorphism of categories
		$$\HomC{\catfont{C}}{C}{U}\cong \HomC{\moplaxn{{\left(\Grothopdiag{M}\right)}\ensuremath{^{\operatorname{op}}}}{\catfont{CAT}}}{W}{\HomC{\catfont{C}}{F(-)}{U}}$$
		$2$-natural in $U\in \catfont{C}$.
	\end{defne}

	\begin{defne}
		Recall from \nameit{Remark}\ref{fromgrothisnotrestrictive} what we can now call \dfn{the trivial marking} and denote as $\opn{triv}$. That is, given $\catfont{A}$ a $2$-category, we can view $\catfont{A}$ as the $2$-$\catfont{Set}$-enriched Grothendieck construction of $\Delta 1\:\catfont{A}\to \catfont{CAT}$ (that produces the identity on $\catfont{A}$ as $2$-$\catfont{Set}$-opfibration). And $\oplaxn$ with respect to the trivial marking coincides with the strict $2$-naturality (see also \nameit{Example}\ref{conicalarelaxnormalconical}).
	\end{defne}

	\begin{rem}
		We can now rephrase \nameit{Theorem}\ref{redoplaxnormalconical} as follows:

		\noindent \textit{In $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ every trivially-marked weighted $2$-colimit can be equivalently expressed as a marked trivially-weighted $2$-colimit.}

		\noindent \textit{More precisely, given $2$-functors $F\:\catfont{A}\to \catfont{C}$ and $W\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to \catfont{CAT}$ with $\catfont{A}$ small,}
		$$\oplaxnmcolim{\opn{triv}}{W}{F}\h[2]\cong\h[2] \oplaxnmcolim{W}{\Delta 1}{\left(F\circ \groth{W}\right)}$$

		So we could say that in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ all the (trivially-marked) weighted $2$-colimits are (now strictly speaking) conicalizable, up to changing the marking.
	\end{rem}

	\begin{exampl}
		Let $F\:\catfont{A}\ensuremath{^{\operatorname{op}}}\to \catfont{CAT}$ be a $2$-functor with $\catfont{A}$ small. Then by \nameit{Example}\ref{univoplaxnormalcoconepresheaves}
		$$F\cong \oplaxnmcolim{F}{\Delta 1}{\left(\yy\circ\h \groth{F}\right)}$$

		In particular, taking $\catfont{A}=1$, we obtain that for every small category $\catfont{D}$
		$$\catfont{D}\cong \oplaxnmcolim{\catfont{D}}{\Delta 1}{\Delta 1}.$$
		Notice that the marking given by $\catfont{D}$ is ``chaotic", in the sense that $\oplaxn$ with respect to it simply becomes $\oplax$.

		So in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ the $2$-functor $\catfont{1}\:\catfont{1}\to \catfont{CAT}$, that we can think of as the $3$-dimensional classifier of $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$, is conically dense (with respect to \nameit{Definition}\ref{defoplaxnormaltwocolimit}), considering the chaotic marking. This completes the idea of \nameit{Remark}\ref{remoneisoplaxnormalconicaldense}.

		Recall that the $1$-dimensional analogue of this, which is that $1\:\catfont{1}\to \catfont{Set}$ (that is the $2$-dimensional classifier of $\catfont{CAT}$) is (conically) dense, was very useful in proving \nameit{Theorem}\ref{copresheavesarekanextensions} (the pointwise Kan extension result for the $\catfont{Set}$-enriched Grothendieck construction).
	\end{exampl}

	\begin{rem}
		We are now ready to propose an original notion of pointwise left Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ along a $2$-$\catfont{Set}$-opfibration, using our definition of colimit in such setting (\nameit{Definition}\ref{defoplaxnormaltwocolimit}). Our idea is to keep the corresponding diagram and weight considered in the (ordinary) enriched setting (see Kelly's~\cite{kelly_basicconceptsofenriched} for the classical definition), but adding the marking that we naturally have when we extend along a $2$-$\catfont{Set}$-opfibration.
	\end{rem}

	\begin{defne}\label{defpointkanextensionlaxthreecat}
		Consider a diagram
		\begin{cd}[7][5]
			\catfont{B} \arrow[rr,"{F}"]\arrow[d,"{K}"']\&\arrow[dl,Rightarrow,shift right=0.75ex,"{\lambda}"{pos=0.355, inner sep= 0.35ex},shorten <=1.65ex,shorten >= 3.35ex]\& \catfont{C} \\
			\catfont{A}\arrow[rru,"L"']
		\end{cd}
		in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ with $\catfont{B}$ small and $K$ a $2$-$\catfont{Set}$-opfibration. Then by \nameit{Theorem}\ref{essentialimagegivenbytwosetopf}, $K$ is isomorphic in the slice $\slice{2\h[2]\mbox{-}\CAT}{\catfont{A}}$ to $\groth{M}$ for some $2$-functor $M\:\catfont{A}\to \catfont{CAT}$. We can assume $K$ is in the form $\groth{M}$, up to whiskering the diagram with the isomorphism in the slice. Assume then that $\lambda$ is a lax normal natural transformation with respect to $M$.

		We say that $\lambda$ exhibits $L$ as the \dfn{pointwise left Kan extension of $F$ along $K$}, written $L=\Lan{K}{F}$, if for every $A\in \catfont{A}$
		$$L(A)\cong \oplaxnopmcolim{M}{\HomC{\catfont{A}}{K(-)}{A}}{F}$$
		with universal oplax normal cocylinder
		\begin{equation}\label{eqpointkanextensiontwocatlax}
			\HomC{\catfont{A}}{K(-)}{A}\aRR{L}\HomC{\catfont{C}}{(L\circ K)(-)}{L(A)}\aRR[\oplaxn]{\HomC{\catfont{C}}{\lambda_{-}}{\id{}}}\HomC{\catfont{C}}{F(-)}{L(A)};
		\end{equation}
		or equivalently if for every $A\in \catfont{A}$ and every $C\in \catfont{C}$ the functor
		$$\HomC{\catfont{C}}{L(A)}{C}\arr{}\HomC{\moplaxn{\catfont{B}\ensuremath{^{\operatorname{op}}}}{\catfont{CAT}}}{\HomC{\catfont{A}}{K(-)}{A}}{\HomC{\catfont{C}}{F(-)}{C}}$$
		given by the oplax normal natural transformation of equation~\refs{eqpointkanextensiontwocatlax} is an isomorphism of categories (notice that the $2$-naturality in $C$ and $A$ is granted, where the latter is using that $L$ is a $2$-functor).
	\end{defne}

	\begin{rem}
		We now prove the pointwise Kan extension result for the $2$-$\catfont{Set}$-enriched Grothendieck construction we were aiming at (from \nameit{Remark}\ref{remaimatpointwise}). Such result is original.
	\end{rem}

	\begin{teor}\label{teorpointkanextforgroth}
		Let $F\:\catfont{A}\to \catfont{CAT}$ be a $2$-functor with $\catfont{A}$ a small $2$-category. Then the $2$-$\catfont{Set}$-enriched Grothendieck construction lax comma object square
		\sq[l][7][7][\lax \opn{comma}][2.7][2.2][0.65]{\Grothop{F}}{\catfont{1}}{\catfont{A}}{\catfont{CAT}}{}{\groth{F}}{\catfont{1}}{F}
		exhibits
		$$F=\Lan{\groth{F}}{\Delta 1}.$$
	\end{teor}
	\begin{proof}
		By \nameit{Theorem}\ref{grothconstrislaxcomma} (and \nameit{Proposition}\ref{laxnatinsidegrothconstr}) we know that the lax natural transformation $\lambda$ that presents the lax comma object is lax normal.

		Given $A\in \catfont{A}$ and $C\in \catfont{CAT}$, we prove that the oplax normal natural transformation
		\begin{equation*}
			\HomC{\catfont{A}}{\groth{F}(-)}{A}\aRR{F}\HomC{\catfont{CAT}}{(F\circ \groth{F})(-)}{F(A)}\aRR[\oplaxn]{\HomC{\catfont{CAT}}{\lambda_{-}}{\id{}}}\HomC{\catfont{CAT}}{\Delta 1(-)}{F(A)},
		\end{equation*}
		that we call $\mu$, is $2$-universal. Explicitly, $\mu$ has components
		\begin{fun}
			\mu_{(B,X)} & \: & \HomC{\catfont{B}}{B}{A} & \longrightarrow & \HomC{\catfont{CAT}}{1}{F(A)} \\[1ex]
			&& \tc*[6]{B}{A}{u}{v}{\theta} &\longmapsto & \tc*[9][30][pos=0.59][pos=0.59][,shift right=0.4ex]{1}{F(A)}{F(u)(X)}{F(v)(X)}{F(\theta)_X}
		\end{fun}
		for every $(B,X)\in \Grothop{F}$ and structure $2$-cells
		$${\left(\mu_{(g,\gamma)}\right)}_u=F(u)(\gamma)\:F(u\circ g)(X')\to F(u)(X)$$
		on every $(g,\gamma)\:(B,X)\al{} (B',X')$ in $\Grothop{F}$, for every $u\:B\to A$ in $\catfont{A}$. Given
		$$\sigma\:\HomC{\catfont{A}}{\groth{F}(-)}{A}\aoplaxn{}\HomC{\catfont{CAT}}{\Delta 1(-)}{C},$$
		we prove that there exists a unique functor $s\:F(A)\to C$ such that
		$$(s\circ -)\circ \mu =\sigma.$$
		We see that there is at most one such $s$, as we need, for every morphism $\alpha\:X\to X'$ in $F(A)$,
		$$s(X)=s\left(\mu_{(A,X)}(\id{A})\right)=\sigma_{(A,X)}(\id{A})$$
		$$s(\alpha)=s\left({\left(\mu_{(\id{A},\alpha)}\right)}_{\id{A}}\right)=\left(\sigma_{(\id{A},\alpha)}\right)_{\id{A}}.$$
		And this $s$ works thanks to the fact that $\sigma$ is oplax normal, since it is readily shown to be functorial (using that $\sigma$ is oplax) and for every $(g,\gamma)\:(B,X)\to (B',X')$ in $\Grothop{F}$ and every $\theta\:u\aR{}v\:B\to A$ in $\catfont{A}$, considering $\underline{u}^X=(u,\id{})$ and $\underline{\theta}^X\:(u,F(\theta)_X)\aR{}(v,\id{})$,
		$$\sigma_{(B,X)}(u)=\sigma_{(A,F(u)(X))}(\id{A})=s(F(u)(X))=s\left(\mu_{(B,X)}(u)\right)$$
		$$\sigma_{(B,X)}(\theta)={\left(\sigma_{(u,F(\theta)_X)}\right)}_{\id{}}={\left(\sigma_{(\id{},F(\theta)_X)}\right)}_{\id{}}=s(F(\theta)_X)=s\left(\mu_{(B,X)}(\theta)\right)$$
		$${\left(\sigma_{(g,\gamma)}\right)}_{u}={\left(\sigma_{(\id{},\gamma)}\right)}_{u}={\left(\sigma_{(u,F(u)(\gamma))}\right)}_{\id{}}={\left(\sigma_{(\id{},F(u)(\gamma))}\right)}_{\id{}}=s(F(u)(\gamma))=s\left({\left(\mu_{(g,\gamma)}\right)}_u\right)$$

		We now prove the $2$-dimensional universality of $\mu$. Given $$\Xi\:\sigma\aM{}\sigma'\:\HomC{\catfont{A}}{\groth{F}(-)}{A}\aoplaxn{}\HomC{\catfont{CAT}}{\Delta 1(-)}{C},$$
		we prove that there exists a unique natural transformation $\xi\:s\aR{} s'\:F(A)\to C$ such that
		$$(\xi\ast -)\h \mu =\Xi.$$
		We see that there is at most one such $\xi$, as we need, for every $X\in F(A)$,
		$$\xi_X=\xi_{\mu_{(A,X)}(\id{A})}=\Xi_{(A,X),\id{A}}$$
		And this $\xi$ works since it is readily shown to be natural (using that $\Xi$ is a modification) and for every $(B,X)\in \Grothop{F}$ and $u\:B\to A$ in $\catfont{A}$
		$$\Xi_{(B,X),u}=\Xi_{(A,F(u)(X)),\id{A}}=\xi_{F(u)(X)}=\xi_{\mu_{(B,X)}(u)}.$$
		We have thus shown that $\mu$ is $2$-universal and this concludes the proof.
	\end{proof}

	\begin{rem}
		The rest of this section is dedicated to the proof that every pointwise left Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ along a $2$-$\catfont{Set}$-opfibration (as defined in \nameit{Definition}\ref{defpointkanextensionlaxthreecat}) is a weak left Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ as well.

		For this, we need a $\oplaxn\mbox{-}\lax$ generalization of the parametrized Yoneda lemma (\nameit{Theorem}\ref{parametrizedyonedaoplaxnlax}) that does not seem to appear in the literature. This lemma will also shed further light on the concept of oplax normal natural transformation. Indeed the idea is the following: a fully lax parametrized Yoneda lemma is not possible, since it is thanks to the strict naturality that we can classically expand the datum on the identity to a complete natural transformation, but our version shows the minimal strictness that we need in order to do so.

		Interestingly, such expansion in the fully strict $2$-natural case classically depends on the naturality of what will be our parameter $A$. Instead, we will need to expand through the slight strictness of the oplax normal naturality in $B$. And an expansion through $B$ is harder to achieve than one through $A$.
	\end{rem}

	\begin{defne}\label{defoplaxnlax}
		Let $G,H\:\catfont{B}\ensuremath{^{\operatorname{op}}}\times \catfont{C}\to \catfont{E}$ be $2$-functors. An \dfn{$\text{\textbf{oplax}}^\text{\textbf{n}}\mbox{-}\text{\textbf{lax}}$ natural transformation} $\alpha$ from $G$ to $H$ is a collection of morphisms
		$$\alpha_{B,C}\:G(B,C)\to H(B,C)$$
		in $\catfont{E}$ for every $(B,C)\in \catfont{B}\ensuremath{^{\operatorname{op}}}\times\catfont{C}$ and, for every $f\:B'\to B$ in $\catfont{B}\ensuremath{^{\operatorname{op}}}$ and $g\:C\to C'$ in $\catfont{C}$, structure $2$-cells
		\begin{eqD*}
			\sq*[o][6][6][\alpha_{f,C}][3.1][3]{G(B',C)}{H(B',C)}{G(B,C)}{H(B,C)}{\alpha_{B',C}}{G(f,\id{})}{H(f,\id{})}{\alpha_{B,C}}\qquad
			\sq*[l][6][6][\alpha_{B,g}][3.1][3]{G(B,C)}{H(B,C)}{G(B,C')}{H(B,C')}{\alpha_{B,C}}{G(\id{},g)}{H(\id{},g)}{\alpha_{B,C'}}
		\end{eqD*}
		such that $\alpha_{-,C}$ is oplax normal natural in $B\in \catfont{B}\ensuremath{^{\operatorname{op}}}$, $\alpha_{B,-}$ is lax natural in $C\in \catfont{C}$ and\v[0.5]
		\begin{eqD*}
			\begin{cd}*[5][-3]
				\&[-1ex] G(B,C)\arrow[rr,"{\alpha_{B,C}}"]\arrow[rd,"{G(\id{},g)}"{inner sep=0.2ex},""'{name=A}]\&[-1ex]\& H(B,C)\arrow[rd,"{H(\id{},g)}"{inner sep=0.2ex}]\arrow[d,Rightarrow,shift right=2.85ex,"{\alpha_{B,g}}"{pos=0.44},shorten <=0.8ex,shorten >=2.2ex]\\
				G(B',C)\arrow[rd,"{G(\id{},g)}"'{inner sep=0.2ex},""{name=B}]\arrow[ru,"{G(f,\id{})}"{inner sep=0.2ex}]\&\&G(B,C')\arrow[rr,"{\alpha_{B,C'}}"]\&\vphantom{.}\arrow[d,Rightarrow,shift right=2.85ex,"{\alpha_{f,C'}}",shorten <=1.05ex,shorten >=1ex]\& H(B,C')\\
				\& G(B',C')\arrow[ru,"{G(f,\id{})}"'{inner sep=0.2ex}]\arrow[rr,"{\alpha_{B',C'}}"']\&\&H(B',C')\arrow[ru,"{H(f,\id{})}"'{inner sep=0.2ex}]
				\arrow[equal,from=A,to=B,shorten <=3.64ex,shorten >=3.64ex]
			\end{cd}\h[2] = \h[2]
			\begin{cd}*[5][-3]
				\& G(B,C)\arrow[rr,"{\alpha_{B,C}}"]\arrow[d,Rightarrow,shift left=2.85ex,"{\alpha_{f,C}}"'{pos=0.44},shorten <=0.8ex,shorten >=2.2ex]\&\&[-1ex] H(B,C)\arrow[rd,"{H(\id{},g)}"{inner sep=0.2ex},""'{name=A}]\\
				G(B',C)\arrow[rr,"{\alpha_{B',C}}"]\arrow[rd,"{G(\id{},g)}"'{inner sep=0.2ex}]\arrow[ru,"{G(f,\id{})}"{inner sep=0.2ex}]\&\vphantom{.}\arrow[d,Rightarrow,shift left=2.85ex,"{\alpha_{B',g}}"',shorten <=1.05ex,shorten >=1ex]\&H(B',C)\arrow[ru,"{H(f,\id{})}"{inner sep=0.2ex}]\arrow[rd,"{H(\id{},g)}"'{inner sep=0.2ex},""{name=B}]\&\&[-1ex] H(B,C')\\
				\& G(B',C')\arrow[rr,"{\alpha_{B',C'}}"']\&\&H(B',C')\v[2]\arrow[ru,"{H(f,\id{})}"'{inner sep=0.2ex}]
				\arrow[equal,from=A,to=B,shorten <=3.64ex,shorten >=3.64ex]
			\end{cd}
		\end{eqD*}
		We call the last axiom the compatibility axiom.

		A \dfn{modification} $\Theta\:\alpha\aM{}\beta\:G\aR[\opln\mbox{-}\lax]{}H$ between $\oplaxn\mbox{-}\lax$ natural transformations is a collection of $2$-cells
		\tc+[7]{G(B,C)}{H(B,C)}{\alpha_{B,C}}{\beta_{B,C}}{\Theta_{B,C}}
		in $\catfont{E}$ that forms both, fixing $C$, a modification $\alpha_{-,C}\aM{}\beta_{-,C}$, and, fixing $B$, a modification $\alpha_{B,-}\aM{}\beta_{B,-}$.
	\end{defne}

	\begin{teor}[The $\oplaxn\mbox{-}\lax$ parametrized Yoneda lemma]\label{parametrizedyonedaoplaxnlax}
		Let $K\:\catfont{B}\to \catfont{A}$ be a $2$-$\catfont{Set}$-opfibration (with small fibres) and $F\:\catfont{B}\ensuremath{^{\operatorname{op}}}\times \catfont{A}\to \catfont{CAT}$ be a $2$-functor. There is a bijection between
		$$\alpha_{B,A}\:\HomC{\catfont{A}}{K(B)}{A}\to F(B,A)$$
		$\oplaxn\mbox{-}\lax$ natural in $(B,A)\in \catfont{B}\ensuremath{^{\operatorname{op}}}\times \catfont{A}$ and
		$$\eta_B\:1\to F(B,K(B))$$
		extraordinary lax natural in $B\in \catfont{B}$ \pteor{see Hirata's~\cite{hirata_notesonlaxends} for a definition of extraordinary lax natural transformations and modifications between them}.

		Moreover this bijection extends to an isomorphism of categories, considering as morphisms of the two categories respectively the modifications between $\oplaxn\mbox{-}\lax$ natural transformations and the modifications between extraordinary lax natural transformations.
	\end{teor}
	\begin{proof}
		By \nameit{Theorem}\ref{essentialimagegivenbytwosetopf}, we can assume that $K$ is in the form $\groth{M}\:\Grothop{M}\to \catfont{A}$ for a $2$-functor $M\:\catfont{A}\ensuremath{^{\operatorname{op}}} \to \catfont{CAT}$. Given
		$$\alpha_{(B,Y),A}\:\HomC{\catfont{A}}{K(B,Y)}{A}\to F((B,Y),A)$$
		$\oplaxn\mbox{-}\lax$ natural in $((B,Y),A)\in {\left(\Grothop{M}\right)}\ensuremath{^{\operatorname{op}}}\times \catfont{A}$ (see \nameit{Definition}\ref{defoplaxnlax}), we construct $\eta_{(B,Y)}$ as the composite
		$$1\ar{\id{B}} \HomC{\catfont{A}}{B}{B}\ar{\alpha_{(B,Y),B}}F((B,Y),B)$$
		Then $\eta_{(B,Y)}$ is extraordinary lax natural in $(B,Y)\in \Grothop{M}$, with structure $2$-cell on $(g,\gamma)\:(B,Y)\to (B',Y')$ in $\Grothop{M}$ given by the pasting
		\begin{eqD}{diagramfromalphatoeta}
			\begin{cd}*[5][5]
				1 \arrow[r,"{\id{B}}"]\arrow[d,"{\id{B'}}"']\& \HomC{\catfont{A}}{B}{B} \arrow[ld,equal,shorten <=5.05ex,shorten >=5.05ex]\arrow[d,"{g\circ -}"] \arrow[r,"{\alpha_{(B,Y),B}}"]\&F((B,Y),B) \arrow[dd,"{F(\id{},g)}"] \arrow[ld,Rightarrow,"{\alpha_{(B,Y),g}}"{pos=0.61},shorten <=4.45ex,shorten >=2.25ex]
				\\
				\HomC{\catfont{A}}{B'}{B'}\arrow[d,"{\alpha_{(B',Y'),B'}}"'] \arrow[r,"{-\circ g}"']\& \HomC{\catfont{A}}{B}{B'}\arrow[rd,"{\alpha_{(B,Y),B'}}"{inner sep =0.2ex}]\arrow[ld,Rightarrow,"{\alpha_{(g,\gamma),B'}}"{pos=0.56},shorten <=3.9ex,shorten >=3.35ex]
				\\
				F((B',Y'),B') \arrow[rr,"{F((g,\gamma),\id{})}"']\&\&F((B,Y),B')
			\end{cd}
		\end{eqD}
		Indeed it is true in general that, given $2$-functors $T,S\:\catfont{C}\ensuremath{^{\operatorname{op}}}\times \catfont{C}\to \catfont{E}$ the composite
		$$J\ar{\iota_C}T(C,C)\ar{\beta_{C,C}}S(C,C)$$
		is extraordinary lax in $C\in \catfont{C}$ if $\iota_C$ is extraordinary lax in $C$ and $\beta_{C,D}$ is $\oplaxn\mbox{-}\lax$ in $(C,D)\in \catfont{C}\ensuremath{^{\operatorname{op}}}\times \catfont{C}$, with structure $2$-cells given by a pasting like that of equation~\refs{diagramfromalphatoeta} (with now a possibly non-identity $2$-cell also in the upper left square). Such structure $2$-cells surely respect the identities, and they also respect the composition by the compability axiom of $\oplaxn\mbox{-}\lax$ (and oplax and lax naturality). The two dimensional axiom is satisfied as well by moving the external $2$-cell through the diagram using the three $2$-dimensional properties that we have. We can then apply this result to $\eta_{(B,Y)}$ since $\id{K(B,Y)}$ is extraordinary natural in $(B,Y)$ and $\alpha_{(B,Y),K(B',Y')}$ is $\oplaxn\mbox{-}\lax$ natural in $((B,Y),(B',Y'))$ (as $\alpha{(B,Y),A}$ is $\oplaxn\mbox{-}\lax$ in $((B,Y),A)$).\v[0.3]

		Now, given $\eta_B\:1\to F((B,Y),B)$ extraordinary lax natural in $(B,Y)\in \Grothop{M}$, we expand it to functors
		$$\alpha_{(B,Y),A}\:\HomC{\catfont{A}}{K(B,Y)}{A}\to F((B,Y),A)$$
		$\oplaxn\mbox{-}\lax$ natural in $((B,Y),A)\in {\left(\Grothop{M}\right)}\ensuremath{^{\operatorname{op}}}\times \catfont{A}$ as follows, using the oplax normal naturality in $(B,Y)$ (that is the only strictness we have). Given $u\:B\to A$ in $\catfont{A}$, considering $\underline{u}^{Y}=(u,\id{})$, the structure $2$-cell $\alpha_{(u,\id{}),A}=\id{}$ will give us a commutative square
		\sq[n][5.5][10]{\HomC{\catfont{A}}{A}{A}}{F((A,M(u)(Y)),A)}{\HomC{\catfont{A}}{B}{A}}{F((B,Y),A)}{\alpha_{(A,M(u)(Y)),A}}{-\circ u}{F((u,\id{}),A)}{\alpha_{(B,Y),A}}
		So, looking at how we constructed $\eta$ from $\alpha$, in order to reach the bijection we want, we define
		$$\alpha_{(B,Y),A}(u)\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} F((u,\id{}),A)\left(\eta_{(A,M(u)(Y))}\right).$$
		Given $\theta\:u\aR{} v\:B\to A$ in $\catfont{A}$, considering
		$$\underline{\theta}^{Y}\:(u,M(\theta)_Y)\aR{}(v,\id{})\:(B,Y)\to (A,M(v)(Y))$$
		and using that $\alpha_{(v,\id{}),A}=\id{}$, we will have by the $2$-dimensional axiom of oplax normal naturality that
		$$\alpha_{(B,Y),A}(\theta)=F(\underline{\theta}^Y,A)_{\h\alpha_{(A,M(v)(Y)),A}(\id{A})}\circ {\left(\alpha_{(u,M(\theta)_Y),A}\right)}_{\id{A}}$$
		So we firstly define the components of the structure $2$-cells that express the oplax normal naturality of $\alpha_{(B,Y),A}$ in $(B,Y)$ and then we will read how to define the action of $\alpha_{(B,Y),A}$ on morphisms $\theta$.

		Looking at the diagram of equation~\refs{diagramfromalphatoeta} applied to $(\id{B},\gamma)\:(B,Y)\to (B,Y')$ in $\Grothop{M}$, we see that, in order to have a bijection between the $\alpha$'s and the $\eta$'s, we need to define
		$${\left(\alpha_{(\id{B},\gamma),B}\right)}_{\id{B}}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \eta_{(\id{B},\gamma)}.$$
		Whence, given arbitrary $(g,\gamma)\:(B,Y)\to (B',Y')$ in $\Grothop{M}$ and $u\:B\to A$ in $\catfont{A}$, since
		$$(u,\id{})\circ (g,\gamma)=(\id{A},M(u)(\gamma))\circ (u\circ g,\id{}),$$ we need to define
		$${\left(\alpha_{(g,\gamma),A}\right)}_u\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} F((u\circ g,\id{}),A)\left(\eta_{(\id{A},M(u)(\gamma))}\right).$$
		And at this point we define, by the argument above,
		$$\alpha_{(B,Y),A}(\theta)\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} F(\underline{\theta}^Y,A)_{\eta_{(A,M(v)(Y))}}\circ F((u,\id{}),A)\left(\eta_{(\id{A},M(\theta)_Y)}\right)$$
		for every $\theta\:u\aR{}v\:B\to A$ in $\catfont{A}$.

		Looking at the diagram of equation~\refs{diagramfromalphatoeta} applied to $(g,\id{})\:(B,Y)\to (B',M(g)(Y))$ in $\Grothop{M}$, we see that, in order to have a bijection between the $\alpha$'s and the $\eta$'s, we need to define
		$${\left(\alpha_{(B,Y),g}\right)}_{\id{B}}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \eta_{(g,\id{})}.$$
		Whence, given an arbitrary $f\:A\to A'$ in $\catfont{A}$ and $u\:K(B,Y)\to A$ in $\catfont{A}$, by the compatibility axiom of $\oplaxn\mbox{-}\lax$ applied to $(u,\id{})\:(B,Y)\to (A,M(u)(Y))$ in $\Grothop{M}$ and $f\:A\to A'$ in $\catfont{A}$, we need to define
		$${\left(\alpha_{(B,Y),f}\right)}_u\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} F((u,\id{}),A')\left(\eta_{(f,\id{})}\right).$$

		Now, we verify that such assignments work. To show that $\alpha_{(B,Y),A}$ is a functor, consider
		$$u\aR{\theta} v\aR{\rho}w\:B\to A$$
		in $\catfont{A}$. Then
		$$\alpha_{(B,Y),A}(\rho\circ \theta)\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} F(\underline{(\rho\circ \theta)}^Y,A)_{\eta_{(A,M(w)(Y))}}\circ F((u,\id{}),A)\left(\eta_{(\id{A},M(\rho\circ\theta)_Y)}\right)$$
		while $\alpha_{(B,Y),A}(\rho)\circ\alpha_{(B,Y),A}(\theta)$ is equal to
		$$\scalebox{0.935}{$F(\underline{\rho}^Y,A)_{\eta_{(A,M(w)(Y))}}\circ F((v,\id{}),A)\left(\eta_{(\id{A},M(\rho)_Y)}\right)\circ F(\underline{\theta}^Y,A)_{\eta_{(A,M(v)(Y))}}\circ F((u,\id{}),A)\left(\eta_{(\id{A},M(\theta)_Y)}\right)$}$$
		By the extraordinary naturality of $\eta$,
		$$\eta_{(\id{A},M(\rho\circ \theta)_Y)}=F((\id{A},M(\theta)_Y),A)\left(\eta_{(\id{A},M(\rho)_Y)}\right)\circ \eta_{(\id{A},M(\theta)_Y)}$$
		And by the uniqueness of the liftings of $2$-cells through $\groth{M}$, we have that
		$$\underline{(\rho\circ \theta)}^Y=\underline{\rho}^Y\circ (\id{A},M(\rho)_Y)\h \underline{\theta}^Y.$$
		So, by $2$-functoriality of $F$, it suffices to prove that
		$$F(\underline{\theta}^Y,A)_{F((\id{A},M(\rho)_Y),A)\left(\eta_{(A,M(w)_Y)}\right)}\circ F((u,M(\theta)_Y),A)\left(\eta_{(\id{A},M(\rho)_Y)}\right)$$
		is equal to
		$$F((v,\id{}),A)\left(\eta_{(\id{A},M(\rho)_Y)}\right)\circ F(\underline{\theta}^Y,A)_{\eta_{(A,M(v)(Y))}}.$$
		But this is true by naturality of $F(\underline{\theta}^Y,A)$ applied to the morphism
		$$\eta_{(\id{A},M(\rho)_Y)}\:\eta_{(A,M(v)(Y))}\to F((\id{A},M(\rho)_Y),A)\left(\eta_{(A,M(w)(Y))}\right).$$
		The fact that ${\left(\alpha_{(B,Y),f}\right)}_u$ is a natural transformation is checked with techniques similar to the above ones, noticing that
		$$\underline{(f\h\theta)}^Y=(f,\id{})\h\underline{\theta}^Y.$$
		Whereas showing that ${\left(\alpha_{(g,\gamma),A}\right)}_u$ is a natural transformation uses that for $(g,\gamma)\:(B,Y)\to (B',Y')$
		$$\underline{(\theta\h g)}^{Y}=\underline{\theta}^{M(g)(Y)}\h (g,\id{})\quad \text{and}\quad \underline{\theta}^{Y'}\h(\id{},\gamma)=(\id{},M(v)(\gamma))\h[1.5]\underline{\theta}^{M(g)(Y)}.$$
		At this point, it is straightforward to check that $\alpha_{(B,Y),A}$ is $\oplaxn\mbox{-}\lax$ in $((B,Y),A)$. And it is immediatly seen that we obtain a bijection between the $\alpha$'s and the $\eta$'s by construction.

		Finally, we extend such bijection to an isomorphism of categories. Given a modification $$\Theta_{(B,Y),A}\:\alpha_{(B,Y),A}\aR{}\beta_{(B,Y),A}\:\HomC{\catfont{A}}{K(B,Y)}{A}\to F((B,Y),A)$$
		between $\oplaxn\mbox{-}\lax$ natural transformations in $((B,Y),A)$, we send it to the modification
		\begin{cd}[6][7]
			1 \arrow[r,"{\id{B}}"]\&[-1ex]\HomC{\catfont{A}}{B}{B}\arrow[r,bend left=27,"{\alpha_{(B,Y),B}}",""'{name=A}]\arrow[r,bend right=27,"{\beta_{(B,Y),B}}"',""{name=B}]\&F((B,Y),B)
			\arrow[Rightarrow,from=A,to=B,shift right=0.5ex,"{\Theta_{(B,Y),B}}"{description,pos=0.46},shorten <=0.15ex,shorten >=0ex]
		\end{cd}
		between extraordinary lax natural transformations (that the latter is such is easily checked). And such assignment is surely functorial.

		Given a modification
		$$\Gamma_{(B,Y)}\:\eta_{(B,Y)}\aR{}{\eta'}_{(B,Y)}\:1\to F((B,Y),B)$$
		between extraordinary lax natural transformations, we construct a corresponding modification $\Theta$. We see that, if we want to reach an isomorphism of categories, we need to define
		$${\left(\Theta_{(B,Y),B}\right)}_{\id{B}}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} \Gamma_{(B,Y)}.$$
		Whence, given an arbitrary $u\:B\to A$ in $\catfont{A}$, since we want $\Theta_{-,A}$ to be a modification, considering $(u,\id{})\:(B,Y)\to (A,M(u)(Y))$ in $\Grothop{M}$, we need to define
		$${\left(\Theta_{(B,Y),A}\right)}_{u}\mathrel{\mathrel{\mathop\ordinarycolon}\mkern-1.2mu=} F((u,\id{}),A)\left(\Gamma_{(A,M(u)(Y))}\right).$$
		It is straightforward to check that $\Theta$ is then a modification between $\oplaxn\mbox{-}\lax$ natural transformations. And such assignment is surely functorial. At this point, it is immediate to see that the two functors are, by construction, inverses of each other, giving the desidered isomorphism of categories.
	\end{proof}

	\begin{rem}
		We are now ready to show that a pointwise left Kan extension in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ along a $2$-$\catfont{Set}$-opfibration is always a weak left Kan extension.
	\end{rem}

	\begin{prop}\label{pointkanisalsoweak}
		Consider a diagram
		\begin{cd}[7][5]
			\catfont{B} \arrow[rr,"{F}"]\arrow[d,"{K}"']\&\arrow[dl,Rightarrow,shift right=0.75ex,"{\lambda}"{pos=0.355, inner sep= 0.35ex},shorten <=1.65ex,shorten >= 3.35ex]\& \catfont{C} \\
			\catfont{A}\arrow[rru,"L"']
		\end{cd}
		in $2\h[2]\mbox{-}\CAT_{\h[-3]\lax}$ with $\catfont{B}$ small and $K$ a $2$-$\catfont{Set}$-opfibration. Assume that $\lambda$ exhibits $L=\Lan{K}{F}$ \pteor{in the sense of \nameit{Definition}\ref{defpointkanextensionlaxthreecat}}. Then $\lambda$ also exhibits $L=\lan{K}{F}$.
	\end{prop}
	\begin{proof}
		Since $L=\Lan{K}{F}$, for every $C$, the $2$-cell
		\begin{equation*}
			\HomC{\catfont{A}}{K(-)}{A}\aRR{L}\HomC{\catfont{C}}{(L\circ K)(-)}{L(A)}\aRR[\oplaxn]{\HomC{\catfont{C}}{\lambda_{-}}{\id{}}}\HomC{\catfont{C}}{F(-)}{L(A)};
		\end{equation*}
		is $2$-universal, giving an isomorphism of categories
		\begin{equation}\label{eqisomorphismpointkanextnelladimcheeweak}
			\HomC{\catfont{C}}{L(A)}{C}\arr{}\HomC{\moplaxn{\catfont{B}\ensuremath{^{\operatorname{op}}}}{\catfont{CAT}}}{\HomC{\catfont{A}}{K(-)}{A}}{\HomC{\catfont{C}}{F(-)}{C}}
		\end{equation}

		We need to prove that, for every $U\in \mlax{\catfont{A}}{\catfont{C}}$, pasting with $\lambda$ gives an isomorphism of categories
		\begin{equation*}
			\HomC{\mlax{\catfont{A}}{\catfont{C}}}{L}{U}\cong \HomC{\mlax{\catfont{B}}{\catfont{C}}}{F}{U\circ K}.
		\end{equation*}
		So consider a lax natural transformation $\varphi\:L\alax{} U$. For every $A\in \catfont{A}$, the component $\varphi_A\:L(A)\to U(A)$ corresponds to an oplax normal natural transformation
		$$\alpha_{-,A}\:{\HomC{\catfont{A}}{K(-)}{A}}\aoplaxn{}{\HomC{\catfont{C}}{F(-)}{U(A)}}$$
		via the isomorphism of equation~\refs{eqisomorphismpointkanextnelladimcheeweak}. And $\varphi_A$ being lax natural in $A\in \catfont{A}$ precisely corresponds to the oplax normal natural transformations $\alpha_{-,A}$ being lax natural in $\catfont{A}$, with structure $2$-cell on $f\:A\to A'$ in $\catfont{A}$ given by the image of $\varphi_f$ through the isomorphism of equation~\refs{eqisomorphismpointkanextnelladimcheeweak}. This means that the lax natural transformations $\varphi$ precisely correspond to functors $\alpha_{B,A}$ $\oplaxn\mbox{-}\lax$ natural in $(B,A)\in \catfont{B}\ensuremath{^{\operatorname{op}}}\times \catfont{A}$.

		\noindent Consider then a modification $\Sigma\:\varphi\aM{}\psi\:L\alax{}U$. The components $\Sigma_A$ with $A\in \catfont{A}$ correspond to modifications $\Theta_{-,A}$ between oplax normal natural transformations $\alpha_{-,A}$ and $\beta_{-,A}$. And the modification axiom for $\Sigma$ corresponds to the modification axiom for $\Theta_{B,-}$ for every fixed $B\in \catfont{B}$. So the modifications $\Sigma$ precisely correspond to modifications $\Theta$ between $\oplaxn\mbox{-}\lax$ natural transformations $\alpha$ and $\beta$. By the functoriality of the isomorphism of equation~\refs{eqisomorphismpointkanextnelladimcheeweak}, we obtain an isomorphism of categories between $\HomC{\mlax{\catfont{A}}{\catfont{C}}}{L}{U}$ and the category of $\oplaxn\mbox{-}\lax$ natural transformations
		$$\alpha_{B,A}\:{\HomC{\catfont{A}}{K(B)}{A}}\aoplaxn{}{\HomC{\catfont{C}}{F(B)}{U(A)}}$$
		in $(B,A)\in \catfont{B}\ensuremath{^{\operatorname{op}}}\times\catfont{A}$ and modifications between them.

		By \nameit{Theorem}\ref{parametrizedyonedaoplaxnlax} (the $\oplaxn\mbox{-}\lax$ parametrized Yoneda lemma), the latter category is then isomorphic to the category of extraordinary lax natural transformations
		$$1\to \HomC{\catfont{C}}{F(B)}{U(K(B))}$$
		in $B\in \catfont{B}$ and modifications between them, which is isomorphic (for example, by Hirata's paper~\cite{hirata_notesonlaxends}) to $\HomC{\mlax{\catfont{B}}{\catfont{C}}}{F}{U\circ K}$. Therefore we have produced an isomorphism of categories
		\begin{equation*}
			\HomC{\mlax{\catfont{A}}{\catfont{C}}}{L}{U}\cong \HomC{\mlax{\catfont{B}}{\catfont{C}}}{F}{U\circ K},
		\end{equation*}
		and we can read that this is given by pasting with $\lambda$.
	\end{proof}

	\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv