text:
\section{Introduction}

Lack and Street introduced wreaths in \cite{LS} as monads in the Eilenberg-Moore category $\EM^M(\K)$ of monads in a 2-category $\K$.
This short and elegant definition of a wreath
when unpacked provides the necessary
data to recover crossed products in different algebraic settings, as it has been showed {\em e.g.} in \cite{LS, BC}.  With the motivation, on the one hand, to understand why
certain structures in the examples obey the laws of a wreath, and on the other to investigate an analogue of a wreath in terms of bimonads,
we introduced biwreaths and biwreath-like objects in \cite{Femic5}. For this aim we first defined bimonads in $\K$ following the line of \cite{Wisb}
(following the line of \cite{McC, Moe}
would require that the 0-cells of $\K$ posses a monoidal structure, which would be a restriction for our purposes). Then we defined biwreaths as bimonads in the
Eilenberg-Moore category $\bEM(\K)$ of bimonads in $\K$. Biwreaths consist of 1-cells $B,F:\A\to\A$ both over the same 0-cell $\A$ in $\K$, together with some 2-cells
satisfying certain axioms. Biwreaths are in particular wreaths, cowreaths, mixed wreaths and mixed cowreaths.
With the notion of biwreath it becomes clear why the structures in some afore-mentioned  examples have the form they have.
Though, in the biwreaths such as we defined them,
some structure 2-cells necessarily turn out to be trivial (some (co)module structures and (co)cycles).
In order to include the non-trivial structures in the study, we introduced biwreath-like objects and mixed biwreath-like objects.
However, from all these constructions only in the {\em canonical biwreath}
(in which the (co)wreath structure 2-cells are images of the embedding 2-functor $\Bimnd(\K)\hookrightarrow\bEM(\K)$ from the 2-category of bimonads in $\K$)
the 1-cell $FB$ turns out to be a bimonad in $\K$. This is the Radford biproduct in 2-categorical terms. A biwreath-like object turns out to be the Sweedler's crossed (co)product,
while a mixed biwreath-like object turns out to be 3-cocycle twisted comodule monad and 3-cycle twisted module comonad, all in the 2-categorical setting.

In the present paper we study structures similar to the previous ones but in which $FB$ is a bimonad.
While in a (left) biwreath $F$ is a left wreath and left cowreath around $B$, once we fix $F$ to be a left wreath around $B$, the only resting possibility for $FB$ to have a bimonad
structure is to consider $B$ as a right cowreath around $F$. For this reason we call our new objects {\em paired wreaths}.
2-categorical Radford biproduct and Sweedler's crossed (co)product, that are a biwreath and a biwreath-like object, respectively, fit also to the setting of a paired wreath.
Paired wreaths represent a 2-categorical ``origin'' of many crossed (bi)products known in algebra. In particular, when $\K=\hat\C$ is the 2-category induced by a braided monoidal category $\C$,
our paired wreaths and the consequent notion of a Hopf datum recover the cross product bialgebras and Hopf data from \cite{BD}. The latter, in turn, generalize
to the setting of braided monoidal categories all cross products known to the date, \cite{Maj5, Maj6, Sch3, BD1}. Broadening of this ``atlas'' of crossed products in a 2-categorical setting
in order to collect some other structures that appeared in the literature in the last years, is a topic of our future research.

As we mentioned, every paired wreath is a Hopf datum. We introduce Yang-Baxter type equations and a sort of naturality for certain distributive laws in 2-categories in order to study
under which conditions a Hopf datum is a paired wreath, \thref{sufficient for bimonad}. Along the way we analyze when a {\em monad Hopf datum} is a wreath, \prref{psi_1,2}, \prref{psi_4}.

Paired wreaths and Hopf data give rise to new definitions of Yetter-Drinfel`d modules, (co)module (co)monads, 2-(co)cycles and (co)cycle twisted (co)actions in a 2-categorical setting.
These new interpretations of the known notions may be used to study classical results in this new setting and in a convenient 2-category $\K$ ({\em e.g.} the one induced by the braided
monoidal category of modules over a commutative ring), being thus a source for new research.

\bigskip

We organized the paper as follows. In Section 2 we set the notation used, we recall some definitions and results from \cite{Femic5} and we introduce $\tau$-bimonads in 2-categories.
Section 3 is devoted to the definition of paired wreaths and to the analysis of which structures underlie them. In Section 4 we define Hopf data collecting some consequent structures of
paired wreaths and study when a Hopf datum is a paired wreath. Here is where we define Yang-Baxter type equations and the mentioned naturality conditions in 2-categories. We also give a
list of examples of the known crossed (bi)products that come out as special cases of paired wreaths. Finally, we indicate the new definitions in a 2-categorical setting of the classically
known objects, as we listed above.

\section{Preliminaries: notation and recalling some basic notions}

We assume the reader is familiar with braided monoidal categories and string diagram notation, to have basic knowledge of 2-categories, (co)monads and (co)wreaths in
2-categories. For reference we recommend \cite{K, Be, Bo, St1, LS}.
Throughout $\K$ will denote a 2-category, $\C$ will denote a (braided) monoidal category with braiding
$\gbeg{2}{1}
\gbr \gnl
\gend$. By $\hat\C$ we will mean the 2-category induced by $\C$. 1-cells in $\hat\C$ are objects of $\C$ and 2-cells in $\hat\C$ are morphisms in $\C$.
From here it is clear that a composition $YX$ of 1-cells $X: \A\to\B$ and $Y: \B\to\E$,
where $\A, \B, \E$ are 0-cells in $\hat\C$, translates to the tensor product $X\ot Y$ in $\C$, that is, the order of 1-cells in the composition is opposite when seen from $\hat\C$
versus when seen from $\C$. A fortiori the composition of 2-cells is opposite, too, this means that the diagrams written in $\hat\C$ describing identities with 2-cells
are left-right symmetric to the corresponding ones in $\C$. To simplify the interpretation of identities we will be working with we are not going to turn the diagrams
from $\hat\C$ to read them in $\C$. Observe that the composition of composable 1-cells in $\K$ gives them a monoidal structure, thus it is justified to use string diagrams to write
identities between 2-cells in $\K$ (which act on composable 1-cells).
Multiplication and unit of a monoid, commultiplication and counit of a comonoid (both in $\C$ and $\K$) we write respectively:
\begin{center}
\begin{tabular}{ccp{0.5cm}cccp{0.5cm}c}
{\footnotesize multiplication} & {\footnotesize unit} & & {\footnotesize comultiplication} & {\footnotesize counit} & & \\
 $\gbeg{2}{2}
\got{1}{} \gnl
\gmu \gnl
\gend$ & $\gbeg{1}{2}
\got{1}{} \gnl
\gu{1} \gnl
\gend$ & & $
\gbeg{2}{2}
\got{2}{} \gnl
\gcmu \gnl
\gend$ &  $
\gbeg{1}{2}
\got{1}{} \gnl
\gcu{1} \gnl
\gend$
\end{tabular}
\end{center}

\medskip

In \cite[Definition 2.3]{Femic5} we defined modules over a monad, comodules over a comonad and bimonads in a 2-category $\K$. In the same paper we used monadic and comonadic
distributive laws in $\K$.
We recall the respective definitions here.

\begin{defn} \delabel{distr}
Let $(\A, T, \mu, \eta)$ be a monad, $(\A, D, \Delta, \Epsilon)$ a comonad and $F:\A\to\A$ a 1-cell in $\K$.
\begin{enumerate}[(a)]
\item A 2-cell $\psi:TF\to FT$ in $\K$ is called a left monadic distributive law if identities \equref{psi laws} hold.
\item A 2-cell $\phi:FD\to DF$ in $\K$ is called a left comonadic distributive law if identities \equref{phi laws} hold.
\end{enumerate}
\vspace{-1,4cm}
\begin{center} \hspace{-0,2cm}
\begin{tabular}{p{7.2cm}p{0cm}p{8cm}}
\begin{equation}\eqlabel{psi laws}
\gbeg{3}{5}
\got{1}{T}\got{1}{T}\got{1}{F}\gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{T}
\gend=
\gbeg{3}{5}
\got{1}{T}\got{1}{T}\got{1}{F}\gnl
\gmu \gcn{1}{1}{1}{0} \gnl
\gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb  \gnl
\gvac{1} \gcl{1} \gcl{1} \gnl
\gvac{1} \gob{1}{F} \gob{1}{T}
\gend;
\quad
\gbeg{2}{5}
\got{3}{F} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{T}
\gend=
\gbeg{3}{5}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\gcl{2} \gcl{2} \gnl
\gob{1}{F} \gob{1}{T}
\gend
\end{equation} & &
\begin{equation}\eqlabel{phi laws}
\gbeg{3}{5}
\got{1}{F} \got{2}{D}\gnl
\gcl{1} \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gnl
\gob{1}{D} \gob{1}{D} \gob{1}{F}
\gend=
\gbeg{3}{5}
\got{2}{F} \got{1}{\hspace{-0,2cm}D}\gnl
\gcn{1}{1}{2}{2} \gcn{1}{1}{2}{2} \gnl
\gvac{1} \hspace{-0,34cm} \glmpt \gnot{\hspace{-0,34cm}\phi} \grmptb \gnl
\gvac{1} \hspace{-0,2cm} \gcmu \gcn{1}{1}{0}{1} \gnl
\gvac{1} \gob{1}{D} \gob{1}{D} \gob{1}{F}
\gend;
\quad
\gbeg{3}{5}
\got{1}{F} \got{1}{D} \gnl
\gcl{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{F}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{D} \gnl
\gcl{1} \gcl{1} \gnl
\gcl{2}  \gcu{1} \gnl
\gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center}
\end{defn}

\begin{defn} \delabel{bimonad} \cite[Definition 2.2]{Femic5}
A 1-cell $B:\A\to\A$ in $\K$ is called a {\em bimonad} if $(\A, B, \mu, \eta)$ is a monad, $(\A, B, \Delta, \Epsilon)$ a comonad and there is a left monadic and comonadic
distributive law $\lambda: BB\to BB$ in $\K$ such that the following conditions are fulfilled:
$
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1}  \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,2cm} \gcu{1} \gnl
\gob{1}{}
\gend
\hspace{1,5cm}
\gbeg{2}{3}
\gu{1}  \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{B}
\gend=
\gbeg{2}{3}
\gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{B} \gob{1}{B}
\gend
\hspace{1,5cm}
\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{1}{}
\gend=
\Id_{id_{\A}}
\hspace{1,5cm}
\gbeg{3}{5}
\got{1}{B} \got{3}{B} \gnl
\gwmu{3} \gnl
\gvac{1} \gcl{1} \gnl
\gwcm{3} \gnl
\gob{1}{B}\gvac{1}\gob{1}{B}
\gend=
\gbeg{4}{5}
\got{1}{B} \got{3}{B} \gnl
\gcl{1} \gwcm{3} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda} \grmptb \gvac{1} \gcl{1} \gnl
\gcl{1} \gwmu{3} \gnl
\gob{1}{B} \gvac{1} \gob{2}{\hspace{-0,34cm}B.}
\gend
$
\end{defn}

\begin{defn} \delabel{(co)modules} \cite[Definition 2.3]{Femic5}
Let $(\A, T, \mu, \eta)$ be a monad and $(\A, D, \Delta, \Epsilon)$ a comonad in $\K$.
\begin{enumerate}[(a)]
\item A 1-cell $F:\B\to\A$ in $\K$ is called a {\em left $T$-module} if there is a 2-cell $\nu: TF\to F$ such that $\nu(\mu\times\Id_F)=\nu(\Id_T\times\nu)$
and $\nu(\eta\times\Id_F)=\Id_F$ holds.
\item A 1-cell $F:\A\to\B$ in $\K$ is called a {\em right $T$-module} if there is a 2-cell $\nu: FT\to F$ such that $\nu(\Id_F\times\mu)=\nu(\nu\times\Id_T)$
and $\nu(\Id_F\times\eta)=\Id_F$ holds.
\item A 1-cell $F:\B\to\A$ in $\K$ is called a {\em left $D$-comodule} if there is a 2-cell $\lambda: F\to DF$ such that $(\Delta\times\Id_F)\lambda=(\Id_D\times\lambda)\lambda$
and $(\Epsilon\times\Id_F)\lambda=\Id_F$ holds.
\item A 1-cell $F:\A\to\B$ in $\K$ is called a {\em right $D$-comodule} if there is a 2-cell $\rho: F\to FD$ such that $(\Id_F\times\Delta)\rho=(\rho\times\Id_D)\rho$
and $(\Id_F\times\Epsilon)\rho=\Id_F$ holds.
\end{enumerate}
\end{defn}

\medskip

In \cite[Proposition 2.4]{Femic5} we proved:

\begin{prop}  \prlabel{distr-actions}

Let $F:\A\to\A$ be a 1-cell in $\K$.
\begin{enumerate}[(a)]
\item
Given a monad  $(\A, B, \mu, \eta)$ with a left monadic distributive law $\psi:BF\to FB$ and a 2-cell
$\Epsilon=
\gbeg{2}{1}
\got{1}{B} \gnl
\gcu{1} \gnl
\gend$ such that
$\gbeg{2}{2}
\got{1}{B} \got{1}{B} \gnl
\gcu{1}  \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,2cm} \gcu{1} \gnl
\gend$ and
$\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{1}{}
\gend=
\Id_{id_{\A}}$
hold, then the 2-cell:
\begin{equation*}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F}
\gend=
\gbeg{2}{4}
\got{1}{B} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F}
\gend
\end{equation*}
makes $F$ a left $B$-module.
\item
Given a comonad $(\A, B, \Delta, \Epsilon)$ with a left comonadic distributive law $\phi:FB\to BF$ and a 2-cell
$\eta=
\gbeg{2}{2}
\gu{1} \gnl
\gob{1}{B}
\gend$ such that
$\gbeg{2}{2}
\gu{1} \gu{1} \gnl
\gob{1}{B} \gob{1}{B}
\gend=
\gbeg{2}{3}
\gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{B} \gob{1}{B}
\gend$
and
$\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gend=
\Id_{id_{\A}}$ hold, then the 2-cell:
\begin{equation*}
\gbeg{2}{3}
\got{3}{F} \gnl
\glcm \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gnl
\gob{1}{B} \gob{1}{F}
\gend
\end{equation*}
makes $F$ a left $B$-comodule.
\item In particular, given a bimonad $(\A, B, \mu, \eta, \Delta, \Epsilon, \lambda)$ and distributive laws $\psi:BF\to FB$ and $\phi:FB\to BF$ as in (a) and (b), the 2-cells
$\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F}
\gend$ and
$\gbeg{2}{3}
\got{3}{F} \gnl
\glcm \gnl
\gob{1}{B} \gob{1}{F}
\gend$ \hspace{0,2cm}
make $F$ a left $B$-module and a left $B$-comodule.
\end{enumerate}
\end{prop}

\subsection{Biwreaths versus biwreath-like objects} \selabel{bw vs bl}

In \cite{Femic5} we defined biwreaths and (mixed) biwreath-like objects, we resume them here.

Firstly, we introduced the 2-category $\bEM(\K)$. It is the Eilenberg-Moore category of bimonads in $\K$. Without entering into details, let us say that
2-cells in $\bEM(\K)$ are pairs $(\rho_M, \rho_C)$, where $\rho_M:F\to GB$ and $\rho_C: FB\to G$ are 2-cells in $\K$ (here $B$ is a 0-cell and $F$ and $G$ come from the involved 1-cells).
We say that $\rho_M$ and $\rho_C$ are {\em canonical} if \equref{canonical} holds. Their {\em canonical restrictions} are given by \equref{restr}. \vspace{-0,4cm}
\begin{center}
\begin{tabular}{p{7,6cm}p{0,6cm}p{6cm}}
\begin{equation} \eqlabel{canonical}
\rho_M=
\gbeg{2}{5}
\got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\rho_M} \grmpb \gnl
\gcl{1} \gcu{1} \gnl
\gcl{1} \gu{1} \gnl
\gob{1}{G} \gob{1}{B} \gnl
\gend,\hspace{0,5cm}
\rho_C=
\gbeg{2}{5}
\got{1}{F} \got{1}{B} \gnl
\gcl{1} \gcu{1} \gnl
\gcl{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\rho_C} \grmp \gnl
\gob{1}{G} \gnl
\gend
\end{equation} &  &
\begin{equation}\eqlabel{restr}
\gbeg{2}{4}
\got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\rho_M} \grmpb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{G}  \gnl
\gend,\hspace{0,5cm}
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\rho_C} \grmp \gnl
\gob{1}{G} \gnl
\gend
\end{equation}
\end{tabular}
\end{center}

A biwreath consists of a pair of 1-cells $B$ and $F$ in $\K$ where $B$ is a bimonad in $\K$ and
$F$ is a bimonad in the Eilenberg-Moore category $\bEM^M(\K)$ over the 0-cell $B$ therein. This means in particular that
$F$ is a wreath (with 2-cells $\mu_M, \eta_M$), cowreath (with 2-cells $\Delta_C, \Epsilon_C$) -- we refer to the latter two as to ``straight structures'' --,
mixed wreath (with 2-cells $\Delta_M, \Epsilon_M$), mixed cowreath (with 2-cells $\mu_C, \eta_C$) -- we refer to the latter two as to ``mixed structures'' -- all around $B$.
Moreover, there is an additional 2-cell in $\bEM(\K)$ (given by 2-cells $\lambda_M, \lambda_C$ in $\K$)
governing the ``monad-comonad'' compatibilities of $F$.
We set the following notation:
$
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gob{2}{F} \gnl
\gend:=
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F} \gnl
\gend \hspace{1,3cm}
\gbeg{1}{4}
\got{1}{} \gnl
\gu{1} \gnl
\gcl{1} \gnl
\gob{1}{F} \gnl
\gend:=
\gbeg{1}{4}
\got{1}{} \gnl
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F} \gnl
\gend \hspace{1,3cm}
\gbeg{2}{3}
\got{2}{F} \gnl
\gcmu \gnl
\gob{1}{F} \gob{1}{F} \gnl
\gend:=
\gbeg{3}{4}
\got{3}{F} \gnl
\glmpb \gnot{\Delta_M} \gcmptb \grmpb \gnl
\gcl{1} \gcl{1} \gcu{1} \gnl
\gob{1}{F} \gob{1}{F} \gnl
\gend \hspace{1,3cm}
\gbeg{1}{3}
\got{1}{F} \gnl
\gcl{1} \gnl
\gcu{1} \gnl
\gob{1}{} \gnl
\gend:=
\gbeg{1}{3}
\got{1}{F} \gnl
\gbmp{\Epsilon_{M}} \gnl
\gcu{1} \gnl
\gob{1}{} \gnl
\gend \hspace{1,3cm}
\sigma:=
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{B} \gnl
\gend
$
and similarly for the comonadic structures.

From the structure of a biwreath it follows that $F$ is a proper left $B$-module and comodule
in $\K$. Moreover, via the (co)unit 2-cells in the Eilenberg-Moore 2-categories (``pre-(co)units'') and the distributive laws $\psi$ and $\phi$ coming from
the wreath and the cowreath structure, we define left and right $F$-action and coaction on $B$ in a broader sense. Namely, the left $F$-(co)module structures on $B$ are
twisted by a so-called 3-(co)cycle (coming from ``mixed structures''), while its right $F$-(co)module structures are twisted by a 2-(co)cycle (coming from ``straight structures'').

From the multiplication-comultiplication compatibility governed by $\lambda$'s (we recall the one for $\lambda_M$ in \equref{8 lambda_M}, this is identity (50) in \cite{Femic5}) we get the following:
when the ``mixed (co)multiplication 2-cells'' $\mu_C$ and $\Delta_M$ are canonical we have
precise non-canonical forms of the ``straight (co)multiplication 2-cells'' $\mu_M$ and $\Delta_C$, respectively
(see \equref{mu_M with sigma}, where we also assumed that $\lambda_M$ is canonical).
Also the other way around: when the straight (co)multiplication 2-cells are canonical, we obtain non-canonical forms of the mixed (co)multiplication 2-cells.
\vspace{-0,6cm}
\begin{center}
\begin{tabular} {p{5.5cm}p{0cm}p{6cm}}
\begin{equation}\eqlabel{8 lambda_M}
\gbeg{4}{6}
\gvac{2} \got{1}{F} \got{1}{F} \gnl
\gvac{2} \glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\glmpb \gnot{\Delta_M} \gcmpb \grmptb \gcl{1} \gnl
\gcl{2} \gcl{2} \gmu \gnl
\gvac{2} \gcn{1}{1}{2}{2} \gnl
\gob{1}{F} \gob{1}{F} \gob{2}{B} \gnl
\gend=
\gbeg{5}{8}
\gvac{1} \got{1}{F} \got{1}{F} \gnl
\gvac{1} \gcl{1} \glmptb \gnot{\Delta_M} \gcmpb \grmpb \gnl
\glmpb \gnot{\lambda_M} \gcmptb \grmptb \gcl{1} \gcl{2} \gnl
\gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{2} \glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gmu \gnl
\gcl{1} \gcl{1} \gcl{1} \gcn{1}{1}{2}{1} \gnl
\gcl{1} \gcl{1}  \gmu \gnl
\gob{1}{F} \gob{1}{F} \gob{2}{B} \gnl
\gend
\end{equation} & & \vspace{0,8cm}
\begin{equation} \eqlabel{mu_M with sigma}
\mu_M=
\gbeg{3}{5}
\got{1}{F} \got{2}{F} \gnl
\gcl{1} \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gcl{1} \gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gob{1}{F} \gob{3}{B} \gnl
\gend\end{equation}
\end{tabular}
\end{center}
\vspace{-0,6cm}
Now, since we consider the (co)unit 2-cells in the Eilenberg-Moore 2-categories canonical, by the above definitions the left and right $F$-(co)module structures
on $B$ must be trivial, and the 2- and 3-(co)cycles are trivial, too.
This means that the proper biwreath will have only left $B$-(co)module structures on $F$. Moreover, since the 2--(co)cycles are trivial, if the ``mixed (co)multiplication 2-cells''
are canonical, so are the ``straight'' ones (see \equref{mu_M with sigma}), and vice versa. Consequently, when all the four latter structure 2-cells of $F$ are canonical, $F$ is a proper monad and comonad.
If $\lambda$'s are canonical, too, $F$ is indeed a bimonad in $\K$.
This follows from the $\Delta_M-\mu_M$ compatibility \equref{8 lambda_M}, which, in the case that all the 2-cells are canonical, becomes:
$$
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gcmu  \gnl
\gob{1}{F}\gob{1}{F}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{2}{F} \gnl
\gcl{1} \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb  \gcl{1} \gnl
\gcl{1} \gmu  \gnl
\gob{1}{F}  \gob{2}{F.}
\gend
$$
Furthermore, the left monadic distributive law $\psi: BF\to FB$ and the left comonadic distributive law $\phi: FB\to BF$ (both with respect to $B$), turn out to be both monadic
and comonadic (at an appropriate side) with respect to $F$. Besides, the pairs of 2-cells $(\psi,\lambda_M)$ and $(\phi,\lambda_C)$ satisfy Yang-Baxter type equations.
Applying (co)unit of $B$ to the afore-mentioned (co)monadic distributive laws of $\psi$ and $\phi$ with respect to $F$, we obtained identities (63--72) in \cite{Femic5},
stating that $F$ is a left $B$-(co)module (co)monad in $\K$.

It is an interesting question which biwreaths one can get if none of these (co)multiplication 2-cells is canonical, we have not investigated yet such examples of biwreaths.

From the above said we may state:

\begin{propdefn}
A biwreath $(B,F)$ in $\K$ for which the following holds:
\begin{itemize}
\item either the mixed or the straight structures are canonical;
\item the canonical restrictions of monadic and comonadic components of the 2-cells coincide;
\end{itemize}
consists of the following data:
\begin{enumerate}
\item left bimonads $B$ and $F$ in $\K$;
\item a left monadic distributive law $\psi: BF\to FB$ and a left comonadic distributive law $\phi: FB\to BF$ (both with respect to $B$), which moreover are monadic
and comonadic (at an appropriate side) with respect to $F$, such that the identities:
$
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\lambda_B} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{B} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_B} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{B}
\gend\hspace{1,4cm}
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gcl{1} \gnl
\gob{1}{F} \gob{1}{F} \gob{1}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gob{1}{F} \gob{1}{F} \gob{1}{B}
\gend\hspace{1,4cm}
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{B} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{F}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{F}
\gend
$
hold, where $\lambda_B:BB\to BB$ and $\lambda_F:FF\to FF$ are left monadic and comonadic distributive laws making $B$ and $F$ left bimonads.
\end{enumerate}

In particular, $F$ is:
\begin{itemize}
\item a left Yetter-Drinfel`d module;
\item a left $B$-module monad;
\item a left $B$-module comonad;
\item a left $B$-comodule monad and
\item a left $B$-comodule comonad
\end{itemize}
in $\K$.

A biwreath described above we will call {\em canonical biwreath}.
\end{propdefn}

The novel notions in a 2-category listed at the end of the above Definition came out as a natural consequence of the notion of a biwreath.

\begin{ex} \exlabel{Example 1}
In \cite[Section 5.1]{Femic5} we proved that a canonical biwreath in $\K=\hat\C$ on a pair of bimonads $B,F$ in $\K$ is given by the 2-cells, {\em i.e.} morphisms in $\C$:
\begin{equation} \eqlabel{lambda Radford}
\psi=
\gbeg{3}{5}
\got{2}{B} \got{1}{F} \gnl
\gcmu \gcl{1} \gnl
\gcl{1} \gbr \gnl
\glm \gcl{1} \gnl
\gob{1}{} \gob{1}{F} \gob{1}{B}
\gend
\hspace{1cm}
\phi=
\gbeg{3}{5}
\got{1}{} \got{1}{F} \got{1}{B} \gnl
\glcm \gcl{1} \gnl
\gcl{1} \gbr \gnl
\gmu \gcl{1} \gnl
\gob{2}{B} \gob{1}{F}
\gend
\hspace{1cm}
\lambda_{Rad}=
\gbeg{4}{7}
\got{1}{} \got{1}{F} \got{3}{F} \gnl
\gwcm{3} \gcl{2} \gnl
\gcl{1} \glcm \gnl
\gcl{1} \gcl{1} \gbr \gnl
\gcl{1} \glm \gcl{2} \gnl
\gwmu{3} \gnl
\gvac{1} \gob{1}{F} \gob{3}{F.}
\gend
\end{equation}
This canonical biwreath recovers the Radford biproduct $B\ot F$ in $\C$.
\end{ex}

As we exposed above, given that in a biwreath the rest of (co)module structures between $B$ and $F$ are trivial, as well as the 2- and 3-(co)cycles,
we introduced biwreath-like objects (consisting of straight structures and an appropriate 2-cell $\lambda_F$) and
mixed biwreath-like objects (consisting of mixed structures and an appropriate 2-cell $\lambda_F$) in \cite{Femic5}. In these new objects the above-mentioned structures are non-trivial.
We recall here the definition of a biwreath-like object ($\EM^M(\K)$ and $\EM^C(\K)$ denote the Eilenberg-Moore categories for monads and comonads, respectively):

\begin{defn} \cite[Definition 6.1]{Femic5} \delabel{bl object}
A biwreath-like object in $\K$ is a monad $(F, \mu_M, \eta_M)$ in $\EM^M(\K)$ and a comonad $(F, \Delta_C, \Epsilon_C)$ in $\EM^C(\K)$ over the same bimonad $B$ in $\K$
with the canonical restrictions:
\begin{equation} \eqlabel{mon comon cell
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gob{2}{F} \gnl
\gend:=
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F} \gnl
\gend \hspace{2cm}
\gbeg{1}{4}
\got{1}{} \gnl
\gu{1} \gnl
\gcl{1} \gnl
\gob{1}{F} \gnl
\gend:=
\gbeg{1}{4}
\got{1}{} \gnl
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F} \gnl
\gend \hspace{2cm}
\gbeg{2}{3}
\got{2}{F} \gnl
\gcmu \gnl
\gob{1}{F} \gob{1}{F} \gnl
\gend:=
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C} \grmptb \gnl
\gob{1}{F} \gob{1}{F} \gnl
\gend \hspace{2cm}
\gbeg{1}{3}
\got{1}{F} \gnl
\gcl{1} \gnl
\gcu{1} \gnl
\gob{1}{} \gnl
\gend:=
\gbeg{1}{3}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\glmpt \gnot{\hspace{-0,34cm}\Epsilon_C} \grmpt \gnl
\gob{1}{} \gnl
\gend
\end{equation}
equipped with a left monadic and comonadic distributive law $\lambda_F:FF\to FF$ with respect to the structure 2-cells \equref{mon comon cell} so that
the following compatibility conditions are fulfilled:
\begin{equation} \eqlabel{lambda mixed 1-3}
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,22cm} \gcu{1} \gnl
\gend \hspace{2cm}
\gbeg{1}{4}
\got{2}{}  \gnl
 \gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{F} \gob{1}{F} \gnl
\gend=
\gbeg{2}{4}
\got{1}{} \gnl
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{F}
\gend  \hspace{2cm}
\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{2}{} \gnl
\gend=
\Id_{id_{\A}}
\end{equation}
and
\begin{center} \hspace{-0,2cm}
\begin{tabular}{p{5.6cm}p{0cm}p{6cm}}
\begin{equation} \eqlabel{lambda_M 8F}
\gbeg{2}{5}
\got{1}{F} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\hspace{-0,22cm} \gcmu \gcn{1}{1}{0}{1} \gnl
\gob{1}{F} \gob{1}{F} \gob{1}{B} \gnl
\gend=
\gbeg{3}{5}
\got{1}{F} \got{2}{F} \gnl
\gcl{1} \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\gob{1}{F} \gob{1}{F} \gob{1}{B} \gnl
\gend
\end{equation} & &
\begin{equation}\eqlabel{lambda_C 8F}
\gbeg{3}{4}
\got{1}{F} \got{1}{F} \got{1}{B} \gnl
\gmu \gcn{1}{1}{1}{0} \gnl
\gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\Delta_C} \grmptb \gnl
\gvac{1} \gob{1}{F} \gob{1}{F} \gnl
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{B} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\Delta_C} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{F.} \gnl
\gend
\end{equation}
\end{tabular}
\end{center}
\end{defn}

Observe that the identity \equref{lambda_M 8F} can be obtained from \equref{8 lambda_M} assuming that $\Delta_M, \lambda_M$ are canonical.

Applying $\Epsilon_B$ to \equref{lambda_M 8F} (or applying $\eta_B$ to \equref{lambda_C 8F}), one gets \equref{bimonad in K F},
then $F$ is a bimonad in $\K$ with possibly non-(co)associative (co)multiplication. On the other hand, applying
$F
\gbeg{1}{2}
\got{1}{F}\gnl
\gcu{1} \gnl
\gend B$ to \equref{lambda_M 8F} and
$F
\gbeg{1}{2}
\gu{1} \gnl
\gob{1}{F}\gnl
\gend B$ to \equref{lambda_C 8F}, we obtain $\mu_M$ and $\Delta_C$ below:  \vspace{-0,7cm}
\begin{center} \hspace{-1,3cm}
\begin{tabular}{p{4cm}p{0cm}p{4cm}p{0cm}p{4cm}}
\begin{equation} \eqlabel{bimonad in K F}
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gcmu \gnl
\gob{1}{F}\gob{1}{F}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{2}{F} \gnl
\gcl{1} \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb  \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F}  \gob{2}{F}
\gend
\end{equation} & &
\begin{equation} \eqlabel{mu_M bl}
\mu_M=
\gbeg{3}{5}
\got{1}{F} \got{2}{F} \gnl
\gcl{1} \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gcl{1} \gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gob{1}{F} \gob{3}{B} \gnl
\gend
\end{equation} & &
\begin{equation} \eqlabel{Delta_C bl}
\Delta_C=
\gbeg{3}{5}
\got{1}{F} \got{3}{B} \gnl
\gcl{1} \glmpb \gnot{\hspace{-0,34cm}\rho} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl \gnl
\gob{1}{F} \gob{2}{F} \gnl
\gend
\end{equation}
\end{tabular}
\end{center}
where
\begin{center} \hspace{1,2cm}
\begin{tabular}{p{4cm}p{1cm}p{4cm}}
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gcl{1} \gnl
\gob{3}{B}
\gend:=
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{B}
\gend
& &
\gbeg{2}{4}
\got{3}{B} \gnl
\gvac{1} \gcl{1} \gnl
\glmpb \gnot{\hspace{-0,34cm}\rho} \grmptb \gnl
\gob{1}{F}\gob{1}{F}
\gend:=
\gbeg{2}{4}
\got{3}{B} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C} \grmptb \gnl
\gob{1}{F}\gob{1}{F.}
\gend
\end{tabular}
\end{center}

\medskip

Now the following is obvious:

\begin{cor}
A canonical biwreath is a biwreath-like object.
\end{cor}

\begin{ex} \exlabel{Example 2}
In \cite[Section 6.1]{Femic5} we proved that in $\K=\hat\C$ a biwreath-like object on a pair of bimonads $B,F$ in $\K$ is given by the 2-cells, {\em i.e.} morphisms in $\C$: \vspace{-0,7cm}
$$\psi=
\gbeg{3}{5}
\got{1}{B} \got{2}{F} \gnl
\gcl{1} \gcmu \gnl
\gbr \gcl{1} \gnl
\gcl{1} \grm \gnl
\gob{1}{F} \gob{1}{B}
\gend
\hspace{1cm}
\phi=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \gnl
\gcl{1} \grcm \gnl
\gbr \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{B} \gob{2}{F}
\gend
\hspace{1cm}
\lambda=
\gbeg{3}{5}
\got{2}{F} \got{1}{F} \gnl
\gcmu \gcl{1} \gnl
\gcl{1} \gbr  \gnl
\gmu \gcl{1} \gnl
\gob{2}{F} \gob{1}{F}
\gend \hspace{0,8cm} \Rightarrow \hspace{0,8cm}
\mu_M=
\gbeg{3}{5}
\got{2}{F} \got{2}{F} \gnl
\gcmu  \gcmu \gnl
\gcl{1} \gbr \gcl{1} \gnl
\gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb  \gnl
\gob{2}{F} \gob{3}{B}
\gend
$$For $\C=R\x\Mod$, the category of modules over a commutative ring $R$, we recover Sweedler's normalized 2-cocycle $\sigma: F\ot F\to B$, twisted $F$-action on $B$ (``$F$ measures $B$'')
and Sweedler's crossed product algebra $F\ot B$.
\end{ex}

\subsection{$\tau$-bimonads}

In \deref{bimonad} we defined left bimonads in $\K$ - the definition involves left monadic and comonadic distributive law $\lambda$. A right bimonad in $\K$ is defined in the
obvious way.

\begin{defn} \delabel{tau-bim}
A monad and a comonad $B$ in $\K$ we call a {\em $\tau$-bimonad} if there is a 2-cell $\tau_{B,B}:BB\to BB$ which is a left and right monadic and comonadic distributive law
such that the following compatibility conditions hold:
$$\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1}  \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,2cm} \gcu{1} \gnl
\gob{1}{}
\gend \hspace{1,6cm}
\gbeg{2}{3}
\gu{1}  \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{B}
\gend=
\gbeg{2}{3}
\gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{B} \gob{1}{B}
\gend \hspace{1,6cm}
\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{1}{}
\gend=
\Id_{id_{\A}}  \hspace{1,6cm}
\gbeg{3}{5}
\got{1}{B} \got{3}{B} \gnl
\gwmu{3} \gnl
\gvac{1} \gcl{1} \gnl
\gwcm{3} \gnl
\gob{1}{B}\gvac{1}\gob{1}{B}
\gend=
\gbeg{2}{5}
\got{2}{B} \got{2}{B} \gnl
\gcmu \gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gmu \gmu \gnl
\gob{2}{B} \gob{2}{B.} \gnl
\gend
$$
\end{defn}

It is straightforward to show that a $\tau$-bimonad $B$ is both a left and a right bimonad with the corresponding 2-cells $\lambda_l$ and $\lambda_r$ being given by:
$$\lambda_l=
\gbeg{4}{5}
\gvac{1} \got{1}{B} \got{3}{B} \gnl
\gwcm{3} \gcl{1} \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gnl
\gwmu{3} \gcl{1} \gnl
\gvac{1} \gob{1}{B} \gvac{1} \gob{2}{\hspace{-0,34cm}B}
\gend \hspace{1cm}\textnormal{and}\hspace{1cm}
\lambda_r=
\gbeg{4}{5}
\got{1}{B} \got{3}{B} \gnl
\gcl{1} \gwcm{3} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gvac{1} \gcl{1} \gnl
\gcl{1} \gwmu{3} \gnl
\gob{1}{B} \gvac{1} \gob{2}{\hspace{-0,34cm}B.}
\gend
$$
respectively.

\subsection{When a biwreath-like object is a $\tau$-bimonad?} \sslabel{when is bimonad}

In a biwreath-like object $(B,F)$ the 1-cell $FB$ is a monad and a comonad by the wreath product and the cowreath coproduct structures, these are given by:
\begin{equation}  \eqlabel{wreath (co)product - old}
\nabla_{FB}=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \got{1}{F} \got{3}{B}  \gnl
\gcl{1}  \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gvac{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gwmu{3} \gnl
\gcl{1} \gwmu{3} \gnl
\gob{1}{F} \gvac{1} \gob{1}{B}
\gend\hspace{1,5cm}
\eta_{FB}=
\gbeg{2}{3}
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{B}
\gend; \hspace{1,5cm}
\Delta_{FB}=
\gbeg{3}{5}
\got{1}{F} \got{3}{B} \gnl
\gcl{1} \gwcm{3} \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C} \grmptb \gwcm{3} \gnl
\gcl{1}  \glmptb \gnot{\hspace{-0,34cm}\phi} \grmptb \gvac{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{B} \got{1}{F} \got{3}{B}
\gend\hspace{1,5cm}
\Epsilon_{FB}=
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\gcl{1} \gcl{1} \gnl
\glmpt \gnot{\hspace{-0,34cm}\Epsilon_C} \grmpt \gnl
\gend
\end{equation}
(see \cite{LS}). Let $\tau_{B,F}, \tau_{F,F}, \tau_{F,F}, \tau_{F,B}$ be left and right monadic and comonadic distributive laws. Here the term ``(co)monadic'' with respect to $F$
is meant with respect to the canonical restrictions of the structure 2-cells of $F$, recall \equref{mon comon cell}.
It is straightforward to show that
\begin{equation}  \eqlabel{tau FB}
\tau_{FB,FB}:=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \got{1}{F} \got{1}{B} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb  \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb  \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb  \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb  \gcl{1} \gnl
\gob{1}{F} \gob{1}{B} \gob{1}{F} \gob{1}{B}
\gend
\end{equation}
is a left and right monadic and comonadic distributive law for $FB$.

In \cite[Theorem 5.3]{Femic5} we proved that in a canonical biwreath in $\K=\hat\C$ (with suitable $\psi, \phi$ and $\lambda_F$) $F\ot B$ is a Radford biproduct
if and only if $F\ot B$ is a bialgebra in $\C$. For the rest of biwreaths and for biwreath-like objects we could not prove that the respective 1-cells $FB$ are bimonads.
In the next section we introduce another concept, which is an alternative definition of a biwreath-like object and it has a structure of a bimonad in $\K$.

\section{Paired wreaths}  \selabel{Paired}

To the (mixed) (co)wreaths that we were dealing with in \cite{Femic5} and were speaking about in \seref{bw vs bl}, we refer to as to {\em left} ones.
Before we proceed, let us list the notation of the structure 2-cells that we are using for left and right (co)wreaths.

\medskip

In a left wreath $F$ around a monad $B$ in $\K$ we have 2-cells $\psi: BF\to FB, \mu_M:FF\to FB, \eta_M:\Id_{\A}\to FB$ so that $\psi$ is a left monadic distributive law and
further 5 axioms are fulfilled (all these will be detailed later on). Then $F$ is a right wreath around a monad $B$ in $\K$ if it is equipped with 2-cells
$\psi': FB\to BF, \mu_M':FF\to BF, \eta_M':\Id_{\A}\to BF$ so that $\psi'$ is a right monadic distributive law and further 5 axioms, left-right symmetric to the latter 5 axioms, are fulfilled.

A left cowreath $F$ around a comonad $B$ in $\K$ is given by 2-cells $\phi: FB\to BF, \Delta_C: FB\to FF, \Epsilon_C: FB\to\Id_{\A}$ where $\phi$ is a left comonadic
distributive law and further 5 axioms are fulfilled. These axioms are up-down symmetric to the axioms of a left wreath around a monad. Lastly, a right
cowreath $F$ around a comonad $B$ in $\K$ is given by 2-cells $\phi': BF\to FB, \Delta_C': BF\to FF, \Epsilon_C': BF\to\Id_{\A}$ so that $\phi'$ is right comonadic distributive law
and 5 further axioms are fulfilled, which are left-right symmetric those of a left cowreath.

\medskip

Our next objective is to study biwreath-like structures in $\K$ that would cover the most general family of the known crossed products and simultaneously are such that the
composit 1-cell $FB$ is a bimonad in $\K$. For the latter, we will consider $FB$ a monad as in \equref{wreath (co)product - old}, that is, $F$ is a left wreath around $B$,
but for the comonad structure we take the only resting possibility different than in \equref{wreath (co)product - old}: that $B$ is a right cowreath around $F$.
Notice that this means that the comonadic diagrams we had in a biwreath-like object will turn into their left-right
symmetric versions and moreover the r\^oles of $B$ and $F$ will be interchanged. Thus, for example, the 2-cell $\Delta_C: FB\to FF$ will turn into $\Delta_C': FB\to BB$, and so on.
This type of symmetry we will call {\em $\alpha$-symmetry} (we will use this term again from \prref{psi_4} on). The symmetry obtained by composing up-down symmetry (rotating
string diagrams by $\pi$) with the $\alpha$-symmetry we will call {\em $\pi$-symmetry}, in accordance with \cite{BD}.

As for generalizing the known crossed products, in particular, for $\K=\hat\C$ we want to cover most general forms of the morphisms $\psi, \phi, \lambda$ of those studied in
\exref{Example 1}, \exref{Example 2}.
In order to prove in \cite[Lemma 5.2]{Femic5} that $\lambda_{Rad}$ from \equref{lambda Radford} satisfies the necessary left monadic and comonadic distributive laws,
we needed to assume that $\mu_M$ is canonical or that $F$ is a trivial left $B$-module,
and that $F$ is a proper left $B$-comodule. In a general biwreath-like structure none of the latter three conditions need be satisfied. For this reason we will drop the
condition that $\lambda$ should be a proper distributive law from the coming definition. Moreover, the condition \equref{8 lambda_M} on $\lambda$
we will substitute by its consequence \equref{mu_M bl} and another identity, which on one hand will yield \equref{bimonad in K F}, but also a concrete form of the 2-cell $\lambda$, on the other hand.
Observe that the following definition is auto $\pi$-symmetric.

\begin{defn} \delabel{paired bl object}
A paired wreath in $\K$ consists of the following data:
\begin{enumerate}
\item a monad $B$ with a 2-cell $\Epsilon_B=
\gbeg{1}{2}
\got{1}{B} \gnl
\gcu{1} \gnl
\gend$ in $\K$ and a comonad $F$ with a 2-cell $\eta_F=
\gbeg{1}{2}
\gu{1} \gnl
\gob{1}{F} \gnl
\gend$ in $\K$, both over the same 0-cell $\A$ in $\K$, satisfying the compatibility conditions:
\begin{equation} \eqlabel{(co)unital}
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,22cm} \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend \hspace{1,5cm}
\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{2}{} \gnl
\gend B=
\Id_{id_{\A}}
\hspace{2cm}
\gbeg{1}{3}
 \gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{F} \gob{1}{F} \gnl
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{F}
\gend  \hspace{1,5cm}
\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{2}{} \gnl
\gend F=
\Id_{id_{\A}}
\end{equation}
\item so that $(F, \psi: BF\to FB, \mu_M, \eta_M)$ is a left wreath around $B$ and $(B, \phi': FB\to BF, \Delta_C', \Epsilon_C')$ is a right cowreath around $F$
with the canonical restrictions:
\begin{equation} \eqlabel{pre-(co)mult}
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gob{2}{F} \gnl
\gend:=
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F} \gnl
\gend \hspace{2cm}
\gbeg{1}{4}
\got{1}{} \gnl
\gu{1} \gnl
\gcl{1} \gnl
\gob{1}{F} \gnl
\gend=
\gbeg{1}{4}
\got{1}{} \gnl
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F} \gnl
\gend \hspace{2cm}
\gbeg{2}{3}
\got{2}{B} \gnl
\gcmu \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend:=
\gbeg{2}{4}
\got{3}{B} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend \hspace{2cm}
\gbeg{1}{3}
\got{1}{B} \gnl
\gcl{1} \gnl
\gcu{1} \gnl
\gob{1}{} \gnl
\gend=
\gbeg{1}{3}
\got{3}{B} \gnl
\gu{1} \gcl{1} \gnl
\glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gnl
\gob{1}{} \gnl
\gend
\end{equation}
where $\gbeg{1}{2}
\got{1}{B} \gnl
\gcu{1} \gnl
\gend$ and $\gbeg{1}{2}
\gu{1} \gnl
\gob{1}{F} \gnl
\gend$ are from the point 1. The above four 2-cells we will call {\em pre-multiplication} and {\em pre-unit} on $F$ and {\em pre-comultiplication} and {\em pre-counit} on $B$.
We set the following notations:
\vspace{-0,6cm}
\begin{center}
\begin{tabular}{p{6cm}p{0cm}p{6cm}}
\begin{equation} \eqlabel{F left B-mod}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F}
\gend:=
\gbeg{2}{4}
\got{1}{B} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F}
\gend
\end{equation} & &
\begin{equation} \eqlabel{B right F-comod}
\gbeg{2}{3}
\got{1}{B} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F}
\gend:=
\gbeg{2}{4}
\got{3}{B} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,6cm}

 \vspace{-0,6cm}
\begin{center}
\begin{tabular}{p{6cm}p{0cm}p{6cm}}
\begin{equation} \eqlabel{right F-mod}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gob{1}{B}
\gend:=
\gbeg{2}{4}
\got{1}{B} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{B}
\gend
\end{equation} & &
\begin{equation} \eqlabel{left B-comod}
\gbeg{2}{3}
\got{3}{F} \gnl
\grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend:=
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center}

\begin{center} \hspace{1,2cm}
\begin{tabular}{p{4cm}p{1cm}p{4cm}}
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gcl{1} \gnl
\gob{3}{B}
\gend:=
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{B}
\gend
& &
\gbeg{2}{4}
\got{3}{F} \gnl
\gvac{1} \gcl{1} \gnl
\glmpb \gnot{\hspace{-0,34cm}\rho'} \grmptb \gnl
\gob{1}{B}\gob{1}{B}
\gend:=
\gbeg{2}{4}
\got{3}{F} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gob{1}{B}\gob{1}{B.}
\gend
\end{tabular}
\end{center}

\item The following compatibilities between the (co)units and pre-(co)multiplications hold:
\begin{equation} \eqlabel{(co)unital mixed}
\gbeg{1}{3}
 \gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend  \hspace{2,5cm}
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,22cm} \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend
\end{equation}
\item The distributive laws $\psi: BF\to FB$ and $\phi': FB\to BF$ from the point 2. should additionally fulfill:
\begin{equation} \eqlabel{psi unital add}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcu{1} \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend
\hspace{3cm}
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend
\end{equation}
and
\begin{equation} \eqlabel{psi and phi concrete}
\psi=
\gbeg{3}{5}
\got{2}{B} \got{2}{F} \gnl
\gcmu \gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\glm \grmo \gcl{1} \gnl
\gob{1}{} \gob{1}{F} \gob{1}{B}
\gend
\hspace{3cm}
\phi'=
\gbeg{3}{5}
\got{1}{} \got{1}{F} \got{1}{B} \gnl
\grmo \gvac{1} \gcl{1} \grcm \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gcl{1} \gnl
\gmu \gmu \gnl
\gob{2}{B} \gob{2}{F}
\gend
\end{equation}
where $\tau_{B,F}, \tau_{F,F}, \tau_{F,F}, \tau_{F,B}$ are left and right monadic and comonadic distributive laws, where the adjectives ``comonadic'' corresponding to $B$
and ``monadic'' corresponding to $F$ are meant with respect to the 2-cells from the point 2.
\item 2-cells $\lambda_B:BB\to BB, \lambda_F:FF\to FF$ in $\K$
so that the following compatibility conditions are fulfilled:
\vspace{-0,7cm}
\begin{center} \hspace{-1,3cm}
\begin{tabular}{p{7cm}p{0cm}p{7cm}}
\begin{equation} \eqlabel{proj B}
\gbeg{3}{5}
\got{2}{B} \got{1}{B} \gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\lambda_B} \grmptb \gnl
\gcl{1} \gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{B}  \gob{1}{B}
\gend
=
\gbeg{5}{9}
\gvac{1} \got{1}{B} \got{5}{B} \gnl
\gcn{1}{1}{3}{3} \gvac{2} \gwcm{3} \gnl
\gwcm{3} \grcm \gcl{4} \gnl
\grcm \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb  \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{3} \gcl{3}  \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gvac{3} \gmu \gcn{1}{1}{1}{0} \gnl
\gcl{1} \gvac{3} \hspace{-0.32cm} \gmu \gnl
\gvac{1} \hspace{-0.24cm} \gob{1}{B} \gob{1}{B} \gvac{2} \gob{1}{B}
\gend
\end{equation} & &
\begin{equation} \eqlabel{proj F}
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb  \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F}  \gob{2}{F}
\gend
=
\gbeg{6}{9}
\gvac{1} \got{2}{F} \gvac{1} \got{2}{F} \got{1}{\hspace{-0,34cm}F} \gnl
\gvac{1} \gcmu \gvac{1} \gcn{1}{3}{2}{2} \gcn{1}{3}{2}{2} \gnl
\gvac{1} \hspace{-0.32cm} \gcn{1}{1}{2}{1} \gcmu \gnl
\gvac{1} \gcl{4} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\ro'} \grmpb \gnl
\gvac{2} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1}  \gnl
\gvac{2} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \glm \gnl
\gvac{2} \glm \gwmu{3} \gnl
\gvac{1} \gwmu{3} \gcn{1}{1}{3}{3} \gnl
\gvac{2} \gob{1}{F} \gvac{2} \gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,7cm}
and
\vspace{-0,7cm}
\begin{center} \hspace{-1,3cm}
\begin{tabular}{p{6cm}p{0cm}p{6cm}}
\begin{equation} \eqlabel{mu_M paired}
\mu_M=
\gbeg{3}{5}
\got{1}{F} \got{2}{F} \gnl
\gcl{1} \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gcl{1} \gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gob{1}{F} \gob{3}{B} \gnl
\gend
\end{equation} & &
\begin{equation} \eqlabel{Delta_C' paired}
\Delta_C'=
\gbeg{3}{5}
\got{1}{F} \got{3}{B} \gnl
\glmpb \gnot{\hspace{-0,34cm}\rho'} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\lambda_B} \grmptb \gnl
\gmu \gcl{1} \gnl
\gob{2}{B} \gob{1}{B} \gnl
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,7cm}
\item $FB$ is a $\tau$-bimonad where $FB$ has the monad and comonad structure from the wreath product and the cowreath coproduct (see point 2.), that is:
\begin{equation}  \eqlabel{wreath (co)product}
\nabla_{FB}=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \got{1}{F} \got{3}{B}  \gnl
\gcl{1}  \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gvac{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gwmu{3} \gnl
\gcl{1} \gwmu{3} \gnl
\gob{1}{F} \gvac{1} \gob{1}{B}
\gend\hspace{1,5cm}
\eta_{FB}=
\gbeg{2}{3}
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{B}
\gend; \hspace{1,5cm}
\Delta_{FB}=
\gbeg{3}{5}
\gvac{2} \got{1}{F} \got{3}{B} \gnl
\gvac{1} \gwcm{3} \gcl{1} \gnl
\gwcm{3} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gob{1}{F} \gvac{1} \gob{1}{B} \got{1}{F} \got{1}{B}
\gend\hspace{1,5cm}
\Epsilon_{FB}=
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\gcl{1} \gcl{1} \gnl
\glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gnl
\gend
\end{equation}
and $\tau_{FB,FB}$ is given by
$$\tau_{FB,FB}=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \got{1}{F} \got{1}{B} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb  \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb  \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb  \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb  \gcl{1} \gnl
\gob{1}{F} \gob{1}{B} \gob{1}{F} \gob{1}{B.}
\gend
$$
\end{enumerate}
\end{defn}

\begin{cor} \colabel{(co)actions}
Let $(B,F)$ be a paired wreath in $\K$. Then the 2-cells \equref{F left B-mod} and \equref{B right F-comod}
make $F$ a left $B$-module and $B$ a right $F$-comodule. Moreover, it holds:
\begin{equation} \eqlabel{(co)mod (co)alg (co)unital}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gvac{1} \gcu{1}
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend
\hspace{2,5cm}
\gbeg{2}{3}
\gu{1} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend
\end{equation}
and
\begin{equation} \eqlabel{right F-mod counits}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gcu{1}
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend
\hspace{2,5cm
\gbeg{2}{3}
\gvac{1} \gu{1} \gnl
\grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend
\end{equation}
\end{cor}

\begin{proof}
The first claim follows by the points 1. and 2. in \deref{paired bl object} and by the part a) and the right hand-side version of the part b) of \prref{distr-actions}.
(The axioms for $\psi$ and $\phi'$ are listed below in \equref{psi laws for B} and \equref{phi' laws for B}, respectively.) The second assertion follows then by \equref{psi unital add},
and the last one follows from \equref{right F-mod} and \equref{left B-comod}, by \equref{psi unital add}.
\qed\end{proof}

We now list the axioms from the point 2. in \deref{paired bl object}, saying that $(F, \psi: BF\to FB, \mu_M, \eta_M)$ is a left wreath around $B$ and that
$(B, \phi': FB\to BF, \Delta_C', \Epsilon_C')$ is a right cowreath around $F$. We have the 2-cells in $\K$:
$$\psi: BF\to FB, \quad \phi': FB\to BF$$
$$\mu_M:FF\to FB, \quad \eta_M:\Id_{\A}\to FB, \quad \Delta_C': FB\to BB, \quad \Epsilon_C': FB\to\Id_{\A}$$
which obey the $\psi$ and $\phi'$ axioms:
\begin{center} \hspace{-0,8cm}
\begin{tabular}{p{7.4cm}p{0cm}p{8cm}}
\begin{equation}\eqlabel{psi laws for B}
\gbeg{3}{5}
\got{1}{B}\got{1}{B}\got{1}{F}\gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend=
\gbeg{3}{5}
\got{1}{B}\got{1}{B}\got{1}{F}\gnl
\gmu \gcn{1}{1}{1}{0} \gnl
\gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb  \gnl
\gvac{1} \gcl{1} \gcl{1} \gnl
\gvac{1} \gob{1}{F} \gob{1}{B}
\gend;
\quad
\gbeg{2}{5}
\got{3}{F} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{B}
\gend=
\gbeg{2}{5}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\gcl{2} \gcl{2} \gnl
\gob{1}{F} \gob{1}{B}
\gend
\end{equation} & &
\begin{equation}\eqlabel{phi' laws for B}
\gbeg{3}{5}
\got{2}{F} \got{1}{B}\gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{F}
\gend=
\gbeg{3}{5}
\got{2}{F} \got{1}{\hspace{-0,2cm}B}\gnl
\gcn{1}{1}{2}{2} \gcn{1}{1}{2}{2} \gnl
\gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gvac{1} \gcn{1}{1}{1}{0} \hspace{-0,2cm} \gcmu \gnl
\gvac{1} \gob{1}{B} \gob{1}{F} \gob{1}{F}
\gend;
\quad
\gbeg{2}{5}
\got{1}{F} \got{1}{B} \gnl
\gcl{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{B}
\gend=
\gbeg{2}{5}
\got{1}{F} \got{1}{B} \gnl
\gcl{1} \gcl{3} \gnl
\gcu{1} \gnl
\gob{3}{B}
\gend
\end{equation}
\end{tabular}
\end{center}
the 2-cell conditions:
\begin{center} \hspace{-1,4cm}
\begin{tabular}{p{6.2cm}p{1cm}p{7cm}}
\begin{equation} \eqlabel{2-cell mu_M}
\gbeg{3}{6}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F}\gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend
\end{equation} &   &
\begin{equation}\eqlabel{2-cell eta_M}
\gbeg{3}{4}
\got{1}{} \got{3}{B}\gnl
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend=
\gbeg{3}{5}
\got{1}{B}\gnl
\gcl{1} \glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend
\end{equation}
\end{tabular}
\end{center}

\begin{center} \hspace{-2,6cm}
\begin{tabular}{p{7.2cm}p{2cm}p{5cm}}
\begin{equation}\eqlabel{2-cell Delta_C'}
\gbeg{3}{5}
\got{2}{F} \got{1}{B} \gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{F}
\gend=
\gbeg{3}{6}
\got{2}{F} \got{1}{B} \gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{F}
\gend
\end{equation} & &
\begin{equation}\eqlabel{2-cell Epsilon_C'}
\gbeg{3}{5}
\got{2}{F} \got{1}{B}\gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmpb \gnot{\hspace{-0,34cm}\phi'} \grmpb \gnl
\glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gcl{1} \gnl
\gob{5}{F}
\gend=
\gbeg{3}{5}
\got{2}{F} \got{1}{B}\gnl
\gcmu \gcl{1} \gnl
\gcl{2} \glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gnl
\gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center}
and the monad and comonad compatibility laws in the corresponding Eilenberg-Moore categories, which translate into the following conditions in $\K$:
\vspace{-0,4cm}
\begin{center} \hspace{0,4cm}
\begin{tabular}{p{5.2cm}p{2cm}p{6.8cm}}
\begin{equation} \eqlabel{monad law mu_M}
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{F}\gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend=
\gbeg{3}{6}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend
\end{equation} &  & \vspace{0,1cm}
\begin{equation}\eqlabel{monad law eta_M}
\gbeg{3}{6}
\got{1}{} \got{3}{F}\gnl
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend=
\gbeg{2}{4}
\got{1}{F}\gnl
\gcl{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{B}
\gend=
\gbeg{3}{5}
\got{1}{F}\gnl
\gcl{1} \glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,4cm}
$$ \textnormal{ \hspace{-1cm} \footnotesize monad law for $\mu_M$}  \hspace{6,5cm}  \textnormal{\footnotesize monad law for $\eta_M$} $$ \vspace{-0,7cm}

\begin{center}
\begin{tabular}{p{6cm}p{2cm}p{6.8cm}}
\begin{equation} \eqlabel{comonad law Delta_C'}
\gbeg{3}{6}
\got{2}{F} \got{1}{B}\gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gob{1}{B} \gob{1}{B}  \gob{1}{B}
\gend=
\gbeg{3}{5}
\got{2}{F} \got{1}{B}\gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{B}  \gob{1}{B}
\gend
\end{equation} &  &
\begin{equation}\eqlabel{comonad law Epsilon_C'}
\gbeg{3}{6}
\got{2}{F} \got{1}{B}\gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gnl
\gob{1}{B}
\gend=
\gbeg{2}{4}
\got{1}{F} \got{1}{B}\gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{B}
\gend=
\gbeg{3}{5}
\got{2}{F} \got{1}{B}\gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gcl{1} \gnl
\gob{5}{B} \gnl
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,5cm}
$$ \textnormal{\hspace{-0,7cm}  \footnotesize comonad law for $\Delta_C'$}  \hspace{6,5cm}  \textnormal{\footnotesize comonad law for $\Epsilon_C'$} $$

\bigskip

\begin{rem}
Observe that we have the following:
\begin{equation} \eqlabel{trivial psi-phi}
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{F}
\gend\stackrel{\equref{2-cell Epsilon_C'}}{=}
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\gcl{2} \gcu{1} \gnl
\gob{1}{F}
\gend,
\hspace{1,2cm}
\gbeg{2}{4}
\got{3}{F} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gob{1}{F} \gob{1}{B}
\gend\stackrel{\equref{psi laws for B}}{=}
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{B}
\gend,
\hspace{1,2cm}
\gbeg{2}{4}
\got{1}{B} \gnl
\gcl{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gob{1}{F} \gob{1}{B}
\gend\stackrel{\equref{2-cell eta_M}}{=}
\gbeg{2}{4}
\got{3}{B} \gnl
\gu{1} \gcl{2} \gnl
\gcl{1} \gnl
\gob{1}{F} \gob{1}{B}
\gend,
\hspace{1,2cm}
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{B}
\gend\stackrel{\equref{phi' laws for B}}{=}
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\gcu{1} \gcl{2} \gnl
\gob{3}{B}
\gend
\end{equation}
where in the first and the third identity we assumed that the 2-cells $\eta_M:\Id_{\A}\to FB $ and $\Epsilon_C': FB\to\Id_{\A}$ are canonical.
Then if we set:
$
\gbeg{2}{3}
\got{1}{F} \got{1}{B} \gnl
\grmf \gcn{1}{1}{-3}{-3} \gnl
\gob{1}{F}
\gend:=
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{F}
\gend
\hspace{1,5cm}
\gbeg{2}{3}
\got{1}{F} \gnl
\gcl{1} \hspace{-0,42cm} \glmf \gnl
\gvac{1} \gob{1}{F} \gob{1}{B}
\gend:=
\gbeg{2}{4}
\got{3}{F} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gob{1}{F} \gob{1}{B}
\gend
\hspace{1,5cm}
\gbeg{2}{3}
\got{3}{B} \gnl
\grmf \gcn{1}{1}{-1}{-1} \gnl
\gob{1}{F} \gob{1}{B}
\gend:=
\gbeg{2}{4}
\got{1}{B} \gnl
\gcl{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gob{1}{F} \gob{1}{B}
\gend
\hspace{1,5cm}
\gbeg{2}{3}
\got{1}{F} \got{1}{B} \gnl
\glmf \gcn{1}{1}{-1}{-1} \gnl
\gob{3}{B}
\gend:=
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{B}
\gend
$
all these (co)module actions turn out to be trivial.
\end{rem}

\subsection{Structures inside a paired wreath}

Similarly as we did in \cite{Femic5}, we will apply
$\gbeg{2}{2}
\got{1}{B}\gnl
\gcu{1} \gnl
\gend$ to the monadic axioms and
$\gbeg{2}{2}
\gu{1} \gnl
\gob{1}{F} \gnl
\gend$ to the comonadic axioms in the above list and then use the pre-counit-multiplication and the pre-unit-comultiplication compatibility relations \equref{(co)unital}, respectively.
In this way we obtain the following identities that hold in a paired wreath:
\begin{center} \hspace{-1,4cm}
\begin{tabular} {p{6.4cm}p{2cm}p{4cm}}
\begin{equation} \eqlabel{mod alg}
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \glm \gnl
\gwmu{3} \gnl
\gob{3}{F}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{3}{F}\gnl
\gcl{1} \gwmu{3} \gnl
\gcn{1}{1}{1}{3} \gvac{1} \gcl{2} \gnl
\gvac{1} \glm \gnl
\gob{5}{F}
\gend
\end{equation} & &  \vspace{-0,9cm}
\begin{equation} \eqlabel{mod alg unity}
\gbeg{3}{5}
\got{1}{B} \gnl
\gcl{1} \gu{1} \gnl
\glm \gnl
\gvac{1} \gcl{1} \gnl
\gob{3}{F}
\gend=
\gbeg{2}{6}
\got{1}{B} \gnl
\gcl{1} \gnl
\gcu{1} \gnl
\gu{1} \gnl
\gcl{1} \gnl
\gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,2cm}
$$ \textnormal{ \footnotesize module monad}  \hspace{4,5cm}  \textnormal{\footnotesize module monad unity} $$ \vspace{-0,7cm}
$$ \textnormal{\footnotesize from 2-cell cond. of $\mu_M$ \equref{2-cell mu_M}} \hspace{4cm} \textnormal{\footnotesize from 2-cell cond. of $\eta_M$ \equref{2-cell eta_M}} $$

\begin{center} \hspace{-1,4cm}
\begin{tabular} {p{6cm}p{2cm}p{4cm}}
\begin{equation}\eqlabel{comod coalg}
\gbeg{3}{6}
\got{3}{B} \gnl
\gvac{1} \gcl{1} \gnl
\gwcm{3} \gnl
\grcm \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{F}
\gend=
\gbeg{3}{6}
\got{3}{B}\gnl
\gvac{1} \gcl{1} \gnl
\gvac{1} \grcm \gnl
\gvac{1} \gcl{1} \gcn{1}{1}{1}{3} \gnl
\gwcm{3} \gcl{1} \gnl
\gob{1}{B} \gvac{1} \gob{1}{B} \gob{1}{F}
\gend
\end{equation} & & \vspace{0,1cm}
\begin{equation}\eqlabel{comod coalg counity}
\gbeg{2}{4}
\got{1}{B} \gnl
\grcm \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{F}
\gend=
\gbeg{1}{4}
\got{1}{B} \gnl
\gcu{1} \gnl
\gu{1} \gnl
\gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center}
\vspace{-0,2cm}
$$ \textnormal{ \footnotesize \hspace{1cm} comodule comonad}  \hspace{4,5cm}  \textnormal{\footnotesize comodule comonad counity} $$ \vspace{-0,7cm}
$$ \textnormal{\footnotesize from 2-cell cond. of $\Delta_C'$ \equref{2-cell Delta_C'}} \hspace{4cm} \textnormal{\footnotesize from 2-cell cond. of $\Epsilon_C'$ \equref{2-cell Epsilon_C'}} $$

\begin{center} \hspace{-1,4cm}
\begin{tabular} {p{7.2cm}p{0cm}p{6.8cm}}
\begin{equation}\eqlabel{weak assoc. mu_M}
\gbeg{4}{4}
\got{1}{F} \got{1}{F} \got{3}{F} \gnl
\gcl{1} \gwmu{3} \gnl
\gwmu{3} \gnl
\gob{3}{F}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gcl{1} \gnl
\gcl{1} \glm \gnl
\gwmu{3} \gnl
\gob{3}{F}
\gend
\end{equation} & &
\begin{equation}\eqlabel{weak unity eta_M}
\gbeg{3}{5}
\got{1}{} \got{3}{F}\gnl
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gcl{1} \gnl
\gcl{1} \glm \gnl
\gwmu{3} \gnl
\gob{3}{F}
\gend=
\gbeg{1}{4}
\got{1}{F}\gnl
\gcl{2} \gnl
\gob{1}{F}
\gend=
\gbeg{2}{4}
\got{1}{F}\gnl
\gcl{1} \gu{1} \gnl
\gmu \gnl
\gob{2}{F}
\gend
\end{equation}
\end{tabular}
\end{center}
\vspace{-0,5cm}
$$ \textnormal{ \footnotesize \hspace{-1,4cm} weak associativity of $\mu_M$}  \hspace{4,5cm}  \textnormal{\footnotesize weak unity $\eta_M$} $$ \vspace{-0,7cm}
$$ \textnormal{\footnotesize from monad law for $\mu_M$ \equref{monad law mu_M}} \hspace{4cm} \textnormal{\footnotesize from monad law for $\eta_M$ \equref{monad law eta_M}} $$

\pagebreak

\begin{center} \hspace{-1,4cm}
\begin{tabular}{p{7cm}p{0cm}p{6.8cm}}
\begin{equation} \eqlabel{weak coass. Delta_C'}
\gbeg{4}{4}
\got{5}{B} \gnl
\gvac{1} \gwcm{3} \gnl
\gwcm{3} \gcl{1} \gnl
\gob{1}{B} \gvac{1} \gob{1}{B} \gob{1}{B} \gnl
\gend=
\gbeg{3}{5}
\got{3}{B} \gnl
\gwcm{3} \gnl
\grcm \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{B} \gnl
\gend
\end{equation} &  &
\begin{equation}\eqlabel{weak counit Epsilon_C'}
\gbeg{3}{5}
\got{3}{B} \gnl
\gwcm{3} \gnl
\grcm \gcl{1} \gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gnl
\gob{1}{B}
\gend=
\gbeg{1}{4}
\got{1}{B} \gnl
\gcl{2} \gnl
\gob{1}{B}
\gend=
\gbeg{2}{4}
\got{2}{B} \gnl
\gcmu \gnl
\gvac{1} \hspace{-0,42cm} \gcu{1} \gcl{1} \gnl
\gob{5}{B} \gnl
\gend
\end{equation}
\end{tabular}
\end{center}
\vspace{-0,5cm}
$$ \textnormal{ \footnotesize \hspace{-1,6cm} weak coassociativity of $\Delta_C'$}  \hspace{5cm}  \textnormal{\footnotesize weak counity $\Epsilon_C'$} $$ \vspace{-0,7cm}
$$ \textnormal{\footnotesize from comonad law for $\Delta_C'$ \equref{comonad law Delta_C'}} \hspace{4cm} \textnormal{\footnotesize from comonad law for $\Epsilon_C'$ \equref{comonad law Epsilon_C'}} $$

Next, we apply $\gbeg{2}{2}
\got{1}{F}\gnl
\gcu{1} \gnl
\gend$ to the $\psi$ axioms and the monadic axioms and $\gbeg{2}{2}
\gu{1} \gnl
\gob{1}{B} \gnl
\gend$ to the $\phi'$ axioms and the comonadic axioms in the list of identities \equref{psi laws for B} to \equref{comonad law Epsilon_C'}. We obtain the following laws
valid in a paired wreath:
 \begin{center} \hspace{-1,5cm}
\begin{tabular}{p{7.2cm}p{0cm}p{6cm}}
\begin{equation}\eqlabel{F mod alg}
\gbeg{3}{5}
\got{1}{B}\got{1}{B}\got{1}{F}\gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\grmo \gcl{1} \gvac{1} \gcl{1} \gnl
\gwmu{3} \gnl
\gob{3}{B}
\gend=
\gbeg{3}{5}
\got{1}{B}\got{1}{B}\got{1}{F}\gnl
\gmu \gcn{1}{1}{1}{0} \gnl
\gvac{1} \hspace{-0,34cm} \grmo \gcl{1} \gnl
\gvac{1} \gcl{1} \gnl
\gvac{1} \gob{1}{B}
\gend
\end{equation} & &
\begin{equation}\eqlabel{F mod alg unit}
\gbeg{2}{5}
\got{3}{F} \gnl
\gu{1} \gcl{1} \gnl
\grmo \gcl{1} \gnl
\gcl{1} \gnl
\gob{1}{B}
\gend=
\gbeg{2}{4}
\got{1}{F} \gnl
\gcu{1} \gnl
\gu{1} \gnl
\gob{1}{B}
\gend
\end{equation}

\\  {\hspace{2cm} \footnotesize module monad} & &  { \hspace{1cm} \footnotesize module monad unity} \\
\multicolumn{2}{c}{{ \hspace{5cm} \footnotesize \quad\qquad from $\psi$ axioms \equref{psi laws for B}}}
\end{tabular}
\end{center}

\begin{center} \hspace{-1,5cm}
\begin{tabular}{p{7.2cm}p{0cm}p{6cm}}
\begin{equation}\eqlabel{B comod coalg}
\gbeg{3}{5}
\got{3}{F}\gnl
\gwcm{3} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{F}
\gend=
\gbeg{3}{4}
\got{3}{F}\gnl
\grmo \gvac{1} \gcl{1} \gnl
\gcn{1}{1}{1}{0} \hspace{-0,2cm} \gcmu \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{F}
\gend
\end{equation} & & \vspace{0,2cm}
\begin{equation}\eqlabel{B comod coalg counit}
\gbeg{2}{4}
\got{3}{F} \gnl
\grmo \gvac{1} \gcl{1}\gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{B}
\gend=
\gbeg{2}{4}
\got{1}{F} \gnl
\gcu{1} \gnl
\gu{1} \gnl
\gob{1}{B}
\gend
\end{equation}

\\  {\hspace{2cm} \footnotesize comodule comonad} & &  { \hspace{1cm} \footnotesize comodule comonad counity} \\
\multicolumn{2}{c}{{ \hspace{5cm} \footnotesize \quad\qquad from $\phi'$ axioms \equref{phi' laws for B}}}
\end{tabular}
\end{center}

\begin{center}
\begin{tabular} {p{6cm}p{1cm}p{6cm}}
\begin{equation} \eqlabel{weak action}
\gbeg{3}{6}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\sigma} \grmpt \gcl{1} \gnl
\gwmu{3} \gnl
\gob{3}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F}\gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\grmo \gcl{1} \gvac{1} \gcl{1} \gnl
\gwmu{3} \gnl
\gob{3}{B}
\gend
\end{equation} & &
\begin{equation} \eqlabel{weak action unity}
\gbeg{3}{4}
\got{1}{} \got{3}{B}\gnl
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gcl{1} \gnl
\gcu{1} \gmu \gnl
\gob{4}{B}
\gend=
\gbeg{3}{5}
\got{1}{B}\gnl
\gcl{1} \glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gnl
\grmo \gcl{1} \gvac{1} \gcl{1} \gnl
\gwmu{3} \gnl
\gob{3}{B}
\gend
\end{equation}
\end{tabular}
\end{center}
\vspace{-0,6cm}
$$ \textnormal{ \footnotesize twisted action}  \hspace{5,5cm}  \textnormal{\footnotesize twisted action unity} $$ \vspace{-0,7cm}
$$ \textnormal{\footnotesize from 2-cell cond. of $\mu_M$ \equref{2-cell mu_M}} \hspace{4cm} \textnormal{\footnotesize from 2-cell cond. of $\eta_M$ \equref{2-cell eta_M}} $$

\begin{center}
\begin{tabular} {p{6cm}p{1.5cm}p{6cm}}
\begin{equation}\eqlabel{weak coaction}
\gbeg{3}{5}
\got{3}{F} \gnl
\gwcm{3} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1}\gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{F}
\gend=
\gbeg{3}{6}
\got{2}{F} \gnl
\gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\rho'} \grmpb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{F}
\gend
\end{equation} & & \vspace{0,1cm}
\begin{equation}\eqlabel{weak coaction counity}
\gbeg{3}{5}
\got{3}{F}\gnl
\gwcm{3} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1}\gnl
\glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gcl{1} \gnl
\gob{5}{F}
\gend=
\gbeg{3}{5}
\got{2}{F}\gnl
\gcmu \gu{1} \gnl
\gcl{2} \glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gnl
\gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,5cm}
$$ \textnormal{ \footnotesize twisted coaction}  \hspace{5,5cm}  \textnormal{\footnotesize twisted coaction counity} $$ \vspace{-0,7cm}
$$ \textnormal{\footnotesize from 2-cell cond. of $\Delta_C'$ \equref{2-cell Delta_C'}} \hspace{4cm} \textnormal{\footnotesize from 2-cell cond. of $\Epsilon_C'$ \equref{2-cell Epsilon_C'}} $$

\begin{center} \hspace{4cm}
\begin{tabular} {p{6cm}p{1cm}p{6.8cm}}
\begin{equation}\eqlabel{2-cocycle condition}
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{F}\gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gvac{1} \gmu \gnl
\gob{4}{B}
\gend=
\gbeg{3}{6}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gvac{1} \gmu \gnl
\gob{4}{B}
\gend
\end{equation} & &
\begin{equation}\eqlabel{normalized 2-cocycle}
\gbeg{3}{6}
\got{1}{} \got{3}{F}\gnl
\glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gvac{1} \gmu \gnl
\gob{4}{B}
\gend=
\gbeg{1}{4}
\got{1}{F}\gnl
\gcu{1} \gnl
\gu{1} \gnl
\gob{1}{B}
\gend=
\gbeg{3}{5}
\got{1}{F}\gnl
\gcl{1} \glmpb \gnot{\hspace{-0,34cm}\eta_M} \grmpb \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gvac{1} \gmu \gnl
\gob{4}{B}
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,5cm}
$$ \textnormal{ \footnotesize 2-cocycle condition}  \hspace{5,5cm}  \textnormal{\footnotesize normalized 2-cocycle} $$ \vspace{-0,7cm}
$$ \textnormal{\footnotesize from monad law for $\mu_M$ \equref{monad law mu_M}} \hspace{4cm} \textnormal{\footnotesize from monad law for $\eta_M$ \equref{monad law eta_M}} $$

\begin{center} \hspace{3.5cm}
\begin{tabular}{p{6cm}p{1.5cm}p{6.8cm}}
\begin{equation} \eqlabel{2-cycle ro'}
\gbeg{3}{6}
\got{2}{F}\gnl
\gcmu \gvac{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\rho'} \grmpb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{B}
\gend=
\gbeg{3}{5}
\got{2}{F}\gnl
\gcmu \gvac{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\rho'} \grmpb \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{B}
\gend
\end{equation} &  &
\begin{equation}\eqlabel{normalized 2-cycle ro'}
\gbeg{3}{6}
\got{2}{F}\gnl
\gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\rho'} \grmpb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gnl
\gob{1}{B}
\gend=
\gbeg{1}{4}
\got{1}{F} \gnl
\gcu{1} \gnl
\gu{1} \gnl
\gob{1}{B}
\gend=
\gbeg{3}{5}
\got{2}{F}\gnl
\gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\rho'} \grmpb \gnl
\glmpt \gnot{\hspace{-0,34cm}\Epsilon_C'} \grmpt \gcl{1} \gnl
\gob{5}{B} \gnl
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,5cm}
$$ \textnormal{ \footnotesize 2-cycle condition for $\rho'$}  \hspace{5,5cm}  \textnormal{\footnotesize normalized 2-cycle $\rho'$} $$ \vspace{-0,7cm}
$$ \textnormal{\footnotesize from comonad law for $\Delta_C'$ \equref{comonad law Delta_C'}} \hspace{4cm} \textnormal{\footnotesize from comonad law for $\Epsilon_C'$ \equref{comonad law Epsilon_C'}} $$

\bigskip

Having seen the above properties in a paired wreath, we may combine them to get the following ones.

\medskip

When we apply $\Epsilon_B$ to the 2-cocycle condition \equref{2-cocycle condition}, we get:
\begin{equation} \eqlabel{epsilon to sigma}
\gbeg{2}{5}
\got{1}{F} \got{1}{\hspace{-0,34cm}F} \got{1}{\hspace{-0,34cm}F} \gnl
\gcl{1} \hspace{-0,22cm} \gmu \gnl
\gvac{1} \hspace{-0,22cm} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{2} \gcu{1} \gnl
\gob{3}{}
\gend=
\gbeg{2}{5}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gcl{1} \gnl
\gcl{1} \glm \gnl
\glmpt \gnot{\sigma} \gcmp \grmptb \gnl
\gvac{2} \gcu{1} \gnl
\gend
\qquad
\stackrel{FF
\gbeg{2}{2}
\gu{1} \gnl
\gob{1}{F}
\gend
}{\Rightarrow}
\gbeg{2}{6}
\got{1}{F} \got{1}{\hspace{-0,34cm}F}  \gnl
\gcl{2} \gcn{1}{1}{0}{0} \hspace{-0,22cm} \gu{1} \gnl
\gvac{1} \gmu \gnl
\gvac{1} \hspace{-0,22cm} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{2} \gcu{1} \gnl
\gob{3}{}
\gend=
\gbeg{2}{5}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gu{1} \gnl
\gcl{1} \glm \gnl
\glmpt \gnot{\sigma} \gcmp \grmptb \gnl
\gvac{2} \gcu{1} \gnl
\gend\hspace{1cm}{\stackrel{\equref{weak unity eta_M}, \equref{mod alg unity}}{\Longleftrightarrow}}\hspace{0,7cm}
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gcu{1} \gnl
\gend=
\gbeg{2}{7}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gcl{1} \gu{1} \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gcu{1} \gnl
\gend\stackrel{*}{\stackrel{\equref{normalized 2-cocycle}}{=}}
\gbeg{2}{5}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,22cm} \gcu{1} \gnl
\gvac{1} \gu{1} \gnl
\gvac{1} \gcu{1} \gnl
\gend
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend
\end{equation}
where at the place * the equality holds since we assume $\eta_M$ to be canonical, and in the last equality we applied the extreme right identities in \equref{(co)unital} and \equref{(co)unital mixed}.
Dually to the above, from the 2-cycle condition \equref{2-cycle ro'} and assuming that $\Epsilon_C'$ is canonical we get:
\begin{equation} \eqlabel{unit to ro'}
\gbeg{2}{3}
\gvac{1} \gu{1} \gnl
\glmpb \gnot{\hspace{-0,34cm}\rho'} \grmptb \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend
\end{equation}

\medskip

We say that a 2-cocycle $\sigma$ and a 2-cycle $\rho'$, respectively, is {\em trivial}, if the following corresponding identity in:
$$\sigma=
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\gcu{1} \gcu{1} \gnl
\gu{1} \gnl
\gob{1}{B}
\gend \hspace{0,1cm}, \quad
\rho'=
\gbeg{2}{4}
\got{1}{F} \gnl
\gcu{1} \gnl
\gu{1} \gu{1} \gnl
\gob{1}{B} \gob{1}{B}
\gend
$$
holds.

\medskip

We now draw some consequences about the 2-cells lambda, from the point 5. of \deref{paired bl object}. When we apply $\Epsilon_B BB$ to \equref{proj B} and
$FF\eta_F$ to \equref{proj F}, we get:
\vspace{-0,7cm}
\begin{center} \hspace{-1,3cm}
\begin{tabular}{p{5cm}p{0cm}p{5cm}}
\begin{equation} \eqlabel{lambda B}
\lambda_B=
\gbeg{4}{7}
\got{1}{B} \got{3}{B} \gnl
\gcl{1} \gwcm{3} \gnl
\gcl{1} \grcm \gcl{3} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \grmo \gcl{1} \gnl
\gcl{1} \gwmu{3} \gnl
\gob{1}{B} \gob{3}{B}
\gend
\end{equation} & &
\begin{equation} \eqlabel{lambda F}
\lambda_F=
\gbeg{4}{7}
\got{1}{} \got{1}{F} \got{3}{F} \gnl
\gwcm{3} \gcl{2} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gnl
\gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{1} \glm \gcl{2} \gnl
\gwmu{3} \gnl
\gvac{1} \gob{1}{F} \gob{3}{F}
\gend
\end{equation}
\end{tabular}
\end{center} \vspace{-0,7cm}
respectively.
In the first equality we used \equref{B comod coalg counit}, \equref{weak counit Epsilon_C'}, left unital distributive law for $\tau_{F,B}$ and
\equref{normalized 2-cocycle}. The second identity follows by $\pi$-symmetry.
Having this, after applying (co)units of $B$ and $F$, respectively, at appropriate places, and by the same properties as above, we further obtain:
$$
\gbeg{3}{4}
\got{1}{B} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_B} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{B}
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gob{2}{B}
\gend,
\hspace{1cm}
\gbeg{2}{4}
\got{3}{B} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_B} \grmptb \gnl
\gob{1}{B} \gob{1}{B}
\gend=
\gbeg{2}{3}
\got{2}{B} \gnl
\gcmu \gnl
\gob{1}{B} \gob{1}{B}
\gend;
\hspace{1,5cm}
\gbeg{3}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F}
\gend=
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gob{2}{F}
\gend,
\hspace{1cm}
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1}\gu{1}  \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gnl
\gob{1}{F} \gob{1}{F}
\gend=
\gbeg{2}{3}
\got{2}{F} \gnl
\gcmu \gnl
\gob{1}{F} \gob{1}{F}
\gend
$$
and having in mind \equref{(co)mod (co)alg (co)unital}--\equref{right F-mod counits}:
$
\gbeg{2}{4}
\got{1}{B} \gnl
\gcl{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_B} \grmptb \gnl
\gob{1}{B} \gob{1}{B}
\gend=
\gbeg{2}{4}
\got{3}{B} \gnl
\gu{1} \gcl{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{B}
\gend,
\hspace{1cm}
\gbeg{2}{4}
\got{1}{B} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_B} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{B}
\gend=
\gbeg{2}{4}
\got{1}{B} \got{1}{B} \gnl
\gcl{1} \gcl{2} \gnl
\gcu{1} \gnl
\gob{3}{B}
\gend;
\hspace{1,5cm}
\gbeg{2}{4}
\got{3}{F} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gnl
\gob{1}{F} \gob{1}{F}
\gend=
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{F}
\gend,
\hspace{1cm}
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{F}
\gend=
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\gcl{2}  \gcu{1} \gnl
\gob{1}{F}
\gend
$
We also may write now $\mu_M$ and $\Delta_C'$ as follows:
\begin{equation} \eqlabel{mu-delta-final}
\mu_M=
\gbeg{5}{7}
\got{1}{} \got{1}{F} \got{4}{F} \gnl
\gwcm{3} \gcmu \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \gcl{3} \gnl
\gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{1} \glm \gcl{1} \gnl
\gwmu{3} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gob{1}{F} \gob{5}{B}
\gend
\hspace{2cm}
\Delta_C'=
\gbeg{5}{7}
\got{1}{F} \got{5}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\rho'} \grmpb \gwcm{3} \gnl
\gcl{1} \gcl{1} \grcm \gcl{3} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \gcl{1} \grmo \gcl{1} \gnl
\gmu \gwmu{3} \gnl
\gob{2}{B}  \gob{3}{B}
\gend
\end{equation}

\bigskip

Finally, let us see what we get from knowing that in a paired wreath $FB$ is a $\tau$-bimonad. Recall that the multiplication $\nabla_{FB}$ and comultiplication
$\Delta_{FB}$ of $FB$ are given by \equref{wreath (co)product}. The first three identities in \deref{tau-bim} applied to $FB$ are already fulfilled by the first four points
from the definition of a paired wreath, concretely, by \equref{(co)unital}, \equref{psi unital add}, \equref{epsilon to sigma} and \equref{unit to ro'}.
If we apply
$\eta_F BF\eta_B, F\eta_B F\eta_B$ and $\eta_F B\eta_F B$
to $\nabla_{FB}$, we obtain: $\psi, \mu_M$
and $\gbeg{3}{3}
\got{1}{} \got{1}{B} \got{1}{B} \gnl
\gu{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend $, respectively. By $\pi$-symmetry, when we apply
$\Epsilon_F BF\Epsilon_F, \Epsilon_F B\Epsilon_FB$ and $F\Epsilon_B F\Epsilon_B$
to $\Delta_{FB}$, we obtain: $\phi', \Delta_C$
and $\gbeg{3}{3}
\got{2}{F} \got{1}{B} \gnl
\gcmu \gcu{1} \gnl
\gob{1}{F} \gob{1}{F}
\gend$, respectively. We apply the $3\times 3$ combinations of these operations to:
\begin{equation}  \eqlabel{paired biwreath biproduct} \hspace{-4cm}
\gbeg{7}{8}
\gvac{2} \got{1}{F} \got{1}{B} \got{1}{F} \got{3}{B}  \gnl
\gvac{2} \gcl{1}  \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gvac{1} \gcl{1} \gnl
\gvac{2} \glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gwmu{3} \gnl
\gvac{2} \gcl{1} \gwmu{3} \gnl
\gvac{1} \gwcm{3} \gcl{1} \gnl
\gwcm{3} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{1} \gnl
\gob{1}{F} \gvac{1} \gob{1}{B} \got{1}{F} \got{1}{B}
\gend=
\gbeg{3}{11}
\gvac{2} \got{1}{F} \gvac{1} \got{1}{B} \gvac{2} \got{1}{F} \got{3}{B}  \gnl
\gvac{1} \gwcm{3} \gcl{1} \gvac{1} \gwcm{3} \gcl{1} \gnl
\gwcm{3} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gwcm{3} \glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{5} \gnl
\gcl{1} \gvac{1} \gcn{1}{2}{1}{-1} \gcl{1} \gcl{1} \gcl{1} \gvac{1} \gcn{1}{1}{1}{-1} \gcl{1}  \gnl
\gcl{1} \gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb  \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gvac{1} \gcn{1}{2}{1}{-1} \gnl
\gcl{1} \gcl{1} \gcn{2}{1}{3}{1}  \gcl{1} \gcl{1} \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gwmu{3} \glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gwmu{3} \gnl
\gcl{1} \gwmu{3} \gvac{1} \gcl{1} \gwmu{3} \gnl
\gob{1}{F} \gvac{1} \gob{1}{B} \gvac{2} \got{1}{F} \got{3}{B}
\gend
\end{equation}
and we obtain 8 identities (one of them yields a trivial identity). Applying respectively the operations:
$$(\Epsilon_F BF\Epsilon_B)(-)(\eta_F BF\eta_B), \qquad (\Epsilon_F B\Epsilon_FB)(-)(\eta_F BF\eta_B), \qquad (F\Epsilon_B F\Epsilon_B)(-)(\eta_F BF\eta_B),$$
$$(\Epsilon_F BF\Epsilon_F)(-)(F\eta_B F\eta_B), \qquad (\Epsilon_F B\Epsilon_FB)(-)(F\eta_B F\eta_B), \qquad (\Epsilon_F BF\Epsilon_F)(-)(\eta_F B\eta_F B), $$
$$(\Epsilon_F B\Epsilon_FB)(-)(\eta_F B\eta_F B), \qquad (F\Epsilon_B F\Epsilon_B)(-)(F\eta_B F\eta_B)$$
we get:
\begin{equation} \eqlabel{1-3}
\gbeg{2}{4}
\got{1}{B} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{6}{7}
\gvac{1} \got{1}{F} \got{5}{B} \gnl
\gwcm{3} \gwcm{3} \gnl
\grcm \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \grmo \gvac{1} \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\grmo \gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \glm \gnl
\gwmu{3} \gwmu{3} \gnl
\gvac{1} \gob{1}{B} \gvac{2} \gob{1}{F}
\gend,
\hspace{1cm}
\gbeg{2}{4}
\got{1}{B} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gob{1}{B} \gob{1}{B}
\gend=
\gbeg{8}{10}
\gvac{1} \got{1}{B} \gvac{3} \got{1}{F} \gnl
\gvac{1} \gcl{1} \gvac{2} \gwcm{3} \gnl
\gwcm{3} \gwcm{3} \glmptb \gnot{\hspace{-0,34cm}\ro'} \grmpb \gnl
\grcm \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{4} \gnl
\gcl{1} \gcl{1} \gcl{1} \gcl{1} \gvac{1} \gcn{1}{1}{1}{-1} \gcn{1}{2}{1}{-1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb  \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gnl
\grmo \gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gwmu{3} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gwmu{3} \gnl
\gvac{1} \gcl{1} \gvac{2} \gwmu{3} \gnl
\gvac{1} \gob{1}{B} \gvac{2} \gob{3}{B}
\gend,
\hspace{1cm}
\gbeg{2}{4}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gvac{1} \hspace{-0,22cm} \gcmu \gnl
\gvac{1} \gob{1}{F} \gob{1}{F}
\gend=
\gbeg{4}{6}
\gvac{1} \got{1}{B} \got{4}{F} \gnl
\gwcm{3} \gcmu \gnl
\grcm \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \glm \gnl
\glm \gwmu{3} \gnl
\gvac{1} \gob{1}{F} \gvac{1} \gob{1}{F}
\gend
\end{equation}

\begin{equation} \eqlabel{4-6}
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{8}{10}
\gvac{2} \got{1}{F} \gvac{3}  \got{1}{F} \gnl
\gvac{1} \gwcm{3} \gvac{2} \gcl{1}  \gnl
\gwcm{3} \glmptb \gnot{\hspace{-0,34cm}\ro'} \grmpb \gwcm{3}  \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \grmo \gvac{1} \gcl{1} \gnl
\gcl{1} \gvac{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb  \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcn{1}{1}{1}{3} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \glm \gnl
\gvac{1} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gmu \gwmu{3} \gnl
\gvac{2} \gcl{1} \gcn{1}{1}{2}{1} \gvac{2} \gcl{2} \gnl
\gvac{2}  \gmu \gnl
\gvac{2}  \gob{2}{B} \gvac{2}  \gob{1}{F}
\gend,
\hspace{0,5cm}
\gbeg{2}{4}
\got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\mu_M} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\Delta_C'} \grmptb \gnl
\gob{1}{B} \gob{1}{B}
\gend=
\gbeg{10}{11}
\gvac{2} \got{1}{F} \gvac{4}  \got{1}{F} \gnl
\gvac{1} \gwcm{3} \gvac{2} \gwcm{3} \gnl
\gwcm{3} \glmptb \gnot{\hspace{-0,34cm}\ro'} \grmpb \gwcm{3} \glmptb \gnot{\hspace{-0,34cm}\ro'} \grmpb \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gcl{4} \gnl
\gcl{1} \gvac{1} \gcl{2} \gcl{1} \gcl{1} \gcl{1} \gvac{1} \gcn{1}{1}{1}{-1} \gcn{1}{2}{1}{-1}  \gnl
\gcl{1} \gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb  \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gnl
\gcn{1}{1}{1}{3} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gvac{1} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gwmu{3} \gnl
\gvac{2} \gcl{1} \gcn{1}{1}{2}{1} \gvac{2} \gwmu{3} \gnl
\gvac{2}  \gmu \gvac{3} \gcl{1} \gnl
\gvac{2}  \gob{2}{B} \gvac{3}  \gob{1}{B}
\gend,
\hspace{0,2cm}
\gbeg{2}{4}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,22cm} \grcm \gnl
\gvac{1} \gob{1}{B} \gob{1}{F}
\gend=
\gbeg{4}{6}
\gvac{1} \got{1}{B} \got{3}{B} \gnl
\gwcm{3} \grcm \gnl
\grcm \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \glm \gnl
\gmu \gwmu{3} \gnl
\gob{2}{F} \gvac{1} \gob{1}{F}
\gend
\end{equation}

\begin{equation} \eqlabel{proj B-bialg}
\gbeg{3}{5}
\got{1}{B} \got{3}{B} \gnl
\gwmu{3} \gnl
\gvac{1} \gcl{1} \gnl
\gwcm{3} \gnl
\gob{1}{B} \gob{3}{B}
\gend=
\gbeg{6}{9}
\gvac{1} \got{1}{B} \got{5}{B} \gnl
\gwcm{3} \gwcm{3} \gnl
\grcm \gcl{1} \grcm \gcl{3} \gnl
\gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gmu \gnl
\gcn{1}{2}{2}{2} \gvac{2} \gcl{1} \gcn{1}{1}{2}{1} \gnl
\gvac{3} \gmu \gnl
\gob{2}{F} \gob{4}{F}
\gend
\stackrel{\equref{proj B}}{=}
\gbeg{3}{5}
\got{2}{B} \got{1}{B} \gnl
\gcmu \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\lambda_B} \grmptb \gnl
\gmu \gcl{1} \gnl
\gob{2}{B}  \gob{1}{B}
\gend\stackrel{\equref{lambda B}}{=}
\gbeg{4}{7}
\gvac{1} \got{1}{B} \got{5}{B} \gnl
\gwcm{3} \gwcm{3} \gnl
\gcl{1} \gvac{1} \gcl{1} \grcm \gcl{3} \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \gvac{1} \gcl{1} \grmo \gcl{1} \gvac{1} \gnl
\gwmu{3} \gwmu{3} \gnl
\gvac{1} \gob{1}{B}  \gob{5}{B}
\gend
\end{equation}

\begin{equation} \eqlabel{proj F-bialg}
\gbeg{3}{5}
\got{1}{F} \got{3}{F} \gnl
\gwmu{3} \gnl
\gvac{1} \gcl{1} \gnl
\gwcm{3} \gnl
\gob{1}{F} \gob{3}{F}
\gend=
\gbeg{6}{9}
\gvac{1} \got{2}{F} \gvac{1} \got{2}{F} \gnl
\gvac{1} \gcmu \gvac{1} \gcn{1}{1}{2}{2} \gnl
\gcn{2}{1}{3}{2} \gcl{1} \gvac{1} \gcmu \gnl
\gcmu \glmptb \gnot{\hspace{-0,34cm}\ro'} \grmpb \gcl{1} \gcl{1} \gnl
\gcl{3} \glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1}  \gnl
\gvac{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \glm \gnl
\gvac{1} \glm \gwmu{3} \gnl
\gwmu{3} \gcn{1}{1}{3}{3} \gnl
\gvac{1} \gob{1}{F} \gvac{2} \gob{1}{F}
\gend
\stackrel{\equref{proj F}}{=}
\gbeg{3}{5}
\got{1}{F} \got{2}{F} \gnl
\gcl{1} \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb  \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F}  \gob{2}{F}
\gend\stackrel{\equref{lambda F}}{=}
\gbeg{4}{7}
\got{1}{} \got{1}{F} \got{5}{F} \gnl
\gwcm{3} \gwcm{3} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \gvac{1} \gcl{3} \gnl
\gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{1} \glm \gcl{1} \gnl
\gwmu{3} \gwmu{3} \gnl
\gvac{1} \gob{1}{F} \gob{5}{F}
\gend
\end{equation}

\section{Hopf data}

We are going to collect the following properties of a paired wreath:
points 1. and 2. of \deref{paired bl object},
the identities: \equref{mod alg} -- \equref{unit to ro'} and \equref{1-3} -- \equref{proj F-bialg} with the corresponding 2-cells, and the thesis of \coref{(co)actions},
and give a name to that collection of data.

\begin{defn}
A Hopf datum in $\K$ consists of the following data:
\begin{enumerate}
\item a monad $(B,
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gob{2}{B}
\gend,
\gbeg{1}{2}
\gu{1} \gnl
\gob{1}{B}
\gend)$, a comonad $(F,
\gbeg{2}{3}
\got{2}{F} \gnl
\gcmu \gnl
\gob{1}{F} \gob{1}{F}
\gend,
\gbeg{1}{2}
\got{1}{F} \gnl
\gcu{1} \gnl
\gend)$ both over the same 0-cell $\A$ in $\K$, with 2-cells
$\gbeg{2}{3}
\got{2}{B} \gnl
\gcmu \gnl
\gob{1}{B} \gob{1}{B}
\gend,
\gbeg{1}{2}
\got{1}{B} \gnl
\gcu{1} \gnl
\gend$ and
$\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gob{2}{F}
\gend,
\gbeg{1}{2}
\gu{1} \gnl
\gob{1}{F} \gnl
\gend$ in $\K$, satisfying the compatibility conditions:
$
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,22cm} \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend, \hspace{0,5cm}
\gbeg{1}{3}
 \gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend, \hspace{0,2cm}
\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{2}{} \gnl
\gend B=
\Id_{id_{\A}},
\hspace{0,7cm}
\gbeg{1}{3}
 \gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{F} \gob{1}{F} \gnl
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{F}
\gend, \hspace{0,2cm}
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,22cm} \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend, \hspace{0,2cm}
\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{2}{} \gnl
\gend F=
\Id_{id_{\A}};
$$
\item further 2-cells:
\begin{equation} \eqlabel{(co)module cells}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F}
\gend, \hspace{0,6cm}
\gbeg{2}{3}
\got{1}{B} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F}
\gend, \hspace{0,6cm}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gob{1}{B}
\gend, \hspace{0,6cm}
\gbeg{2}{3}
\got{3}{F} \gnl
\grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend
\end{equation}
so that
$\gbeg{2}{1}
\glm \gnl
\gend$ makes $F$ a proper left $B$-module and
$\gbeg{2}{1}
\grcm \gnl
\gend$ makes $B$ a proper right $F$-comodule, and the following relations are fulfilled:
$$
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gvac{1} \gcu{1}
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend,
\hspace{0,8cm}
\gbeg{2}{3}
\gu{1} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend,
\hspace{1,4cm}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gcu{1}
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend,
\hspace{0,8cm}
\gbeg{2}{3}
\gvac{1} \gu{1} \gnl
\grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend
$$
\item 2-cells:
$$
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gob{3}{B}
\gend \hspace{0,2cm}, \hspace{0,6cm}
\gbeg{3}{3}
\got{3}{F} \gnl
\glmpb \gnot{\hspace{-0,34cm}\rho'} \grmptb \gnl
\gob{1}{B}\gob{1}{B}
\gend
$$
and
$$\tau_{B,F}:BF\to FB, \tau_{F,B}:FB\to BF, \tau_{B,B}:BB\to BB, \tau_{F,F}:FF\to FF$$
where $\tau_{B,F}, \tau_{F,F}, \tau_{F,F}, \tau_{F,B}$ are left and right monadic and comonadic distributive laws, where the adjectives ``comonadic'' corresponding to $B$
and ``monadic'' corresponding to $F$ are meant with respect to the 2-cells from the point 1.;
\item so that setting:
$$
\psi=
\gbeg{4}{5}
\got{2}{B} \got{2}{F} \gnl
\gcmu \gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\glm \grmo \gcl{1} \gnl
\gob{1}{} \gob{1}{F} \gob{1}{B}
\gend,
\hspace{0,7cm}
\phi'=
\gbeg{4}{5}
\got{1}{} \got{1}{F} \got{1}{B} \gnl
\grmo \gvac{1} \gcl{1} \grcm \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gcl{1} \gnl
\gmu \gmu \gnl
\gob{2}{B} \gob{2}{F}
\gend,
\hspace{0,7cm}
\mu_M=
\gbeg{5}{7}
\got{1}{} \got{1}{F} \got{4}{F} \gnl
\gwcm{3} \gcmu \gnl
\gcl{1} \glcm \gcl{1} \gcl{3} \gnl
\gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{1} \glm \gcl{1} \gnl
\gwmu{3} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gob{1}{F} \gob{5}{B}
\gend,
\hspace{1cm}
\Delta_C'=
\gbeg{5}{7}
\got{1}{F} \got{5}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\rho'} \grmpb \gwcm{3} \gnl
\gcl{1} \gcl{1} \grcm \gcl{3} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \gcl{1} \grm \gnl
\gmu \gwmu{3} \gnl
\gob{2}{B}  \gob{3}{B}
\gend
$$
the identities \equref{mod alg} -- \equref{unit to ro'} and \equref{1-3} -- \equref{proj F-bialg} hold true.
\end{enumerate}
\end{defn}

\medskip

\begin{cor} \colabel{Hd conseq}
In a Hopf datum the following identities hold true:
$$
\gbeg{2}{4}
\got{1}{B} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{F}
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F}
\gend
\hspace{1,5cm}
\gbeg{2}{4}
\got{1}{B} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcu{1} \gcl{1} \gnl
\gob{3}{B}
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gob{1}{B}
\gend
\hspace{1,5cm}
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{2}{3}
\got{3}{F} \gnl
\grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend
\hspace{1,5cm}
\gbeg{2}{4}
\got{3}{B} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gob{1}{B} \gob{1}{F}
\gend=
\gbeg{2}{3}
\got{1}{B} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F}
\gend
$$

$$
\gbeg{2}{4}
\got{3}{F} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gob{1}{F} \gob{1}{B}
\gend=
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{B}
\gend
\hspace{1,5cm}
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\phi'} \grmptb \gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{B}
\gend=
\gbeg{2}{4}
\got{1}{F} \got{1}{B} \gnl
\gcu{1} \gcl{2} \gnl
\gob{3}{B}
\gend
$$
\end{cor}

\begin{proof}
The first four identities follow by the point 2 of the Definition and relations \equref{weak counit Epsilon_C'} and \equref{weak unity eta_M}, the last two follow by
the point 1 of the Definition and relations \equref{F mod alg unit} and \equref{B comod coalg counit}.
\qed\end{proof}

\begin{cor} \colabel{Hopf data}
Every paired wreath is a Hopf datum.
\end{cor}

\begin{rem} \rmlabel{when cocycles are trivial}
Let us record some properties of a Hopf datum.
Observe that the identities \equref{mod alg}-\equref{comod coalg counity} are $\alpha$-symmetric to the identities \equref{F mod alg}-\equref{B comod coalg counit}, and
that the first identity in \equref{1-3} is auto $\alpha$- and $\pi$-symmetric. Moreover, if $\sigma$ and $\rho'$ are trivial, the 2-cells $\mu_M$ and $\Delta_C'$ are canonical
and we have:
\begin{itemize}
\item the identities \equref{2-cocycle condition}--\equref{unit to ro'} trivially hold,
\item the identities \equref{weak action}--\equref{weak coaction counity} say that the respective (co)actions (\equref{right F-mod}, \equref{left B-comod}) are proper,
which is $\alpha$-symmetric to the fact that the (co)actions \equref{F left B-mod}, \equref{B right F-comod} are proper (\coref{(co)actions}),
\item the identities \equref{weak assoc. mu_M}--\equref{weak counit Epsilon_C'} say that $F$ is a monad and $B$ is a comonad
(this is $\alpha$-symmetric to $B$ being a monad and $F$ a comonad),
\item the second identity in \equref{4-6} holds trivially,
\item the second identity in \equref{1-3} is $\alpha$-symmetric to the third one therein,
and the first identity in \equref{4-6} is $\alpha$-symmetric to the third one therein,
\item the identity \equref{proj B-bialg} is $\alpha$-symmetric to \equref{proj F-bialg}.
\end{itemize}
We have already seen that the definition of a Hopf datum is auto $\pi$-symmetric, now we observe that when $\sigma$ and $\rho'$ are trivial
a Hopf datum is also auto $\alpha$-symmetric.
\end{rem}

\medskip

Observe that in a paired wreath and a Hopf datum some of the (co)module structures \equref{F left B-mod} -- \equref{left B-comod} can be trivial.
The same holds for the (co)cycles $\sigma$ and $\rho'$, as we commented in the above Remark. This gives $2^6$ combinations of structures and thus $2^6$
types of paired wreaths and of Hopf data. Some of them have been studied classically in $\K=\hat Vec$ where $Vec$ is the category of vector spaces over a field $k$ (or modules over a commutative ring).
The rest would yield new structures, not only of crossed products in $Vec$, but even in general $\K$.

\begin{ex}
Let $\K=\hat\C$ where $\C$ is a braided monoidal category with braiding
$\gbeg{2}{1}
\gbr \gnl
\gend$. In this setting our Hopf datum is the Hopf datum defined in \cite[Section 4]{BD}. For trivial $\sigma$ and $\rho'$ one has the Hopf datum from \cite[Section 2]{BD1}.
On the other hand, a normalized cross product bialgebra from \cite[Definition 3.5]{BD} differs from our paired wreath object in ``projection relations'' (3.7), which in our
setting appear substituted by more general conditions \equref{proj B}, \equref{proj F}. Namely, one of the two symmetric projection relations is:
$$\gbeg{4}{4}
\got{2}{B} \got{1}{F} \gnl
\gcmu \gcl{1} \gnl
\gcl{1} \grmo \gcl{1} \gnl
\gob{1}{B} \gob{1}{B}
\gend=
\gbeg{4}{6}
\got{3}{B} \got{1}{F} \gnl
\gwcm{3} \gcl{1} \gnl
\grcm \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gcl{1} \gvac{1} \gmu \gnl
\gob{1}{B} \gob{4}{B}
\gend
$$
Tensoring on the left by $B$ and applying braiding above and below and naturality, this implies:
$$\gbeg{4}{5}
\got{2}{B} \got{1}{B} \got{1}{F} \gnl
\gcmu \gcl{1} \gcl{1} \gnl
\gcl{1} \gbr \gcl{1} \gnl
\gcl{1} \gcl{1} \grmo \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{B}
\gend=
\gbeg{4}{7}
\got{3}{B} \got{1}{B}\got{1}{F} \gnl
\gwcm{3} \gcl{1} \gcl{1} \gnl
\grcm \gbr \gcl{1} \gnl
\gcl{1} \gbr \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{1} \gcl{2} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gcl{1} \gvac{2} \gmu \gnl
\gob{1}{B} \gob{1}{B} \gob{4}{B}
\gend
\qquad\textnormal{which further implies:}\quad
\gbeg{5}{7}
\got{2}{B} \got{3}{B} \gnl
\gcn{2}{1}{2}{2} \gwcm{3} \gnl
\gcmu \grcm \gcl{3} \gnl
\gcl{1} \gbr \gcl{1} \gnl
\gcl{1} \gcl{1} \grmo \gcl{1} \gnl
\gcl{1} \gcl{1} \gwmu{3} \gnl
\gob{1}{B} \gob{1}{B} \gob{3}{B}
\gend=
\gbeg{5}{9}
\gvac{1} \got{1}{B} \got{5}{B} \gnl
\gcn{1}{1}{3}{3} \gvac{2} \gwcm{3} \gnl
\gwcm{3} \grcm \gcl{4} \gnl
\grcm \gbr \gcl{1} \gnl
\gcl{1} \gbr  \glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gcl{3} \gcl{3}  \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gvac{3} \gmu \gcn{1}{1}{1}{0} \gnl
\gcl{1} \gvac{3} \hspace{-0.32cm} \gmu \gnl
\gvac{1} \hspace{-0.24cm} \gob{1}{B} \gob{1}{B} \gvac{2} \gob{1}{B}
\gend
$$
The latter is our \equref{proj B} with $\tau$'s corresponding to the braiding. The argument for the other identity is similar.
\end{ex}

In \cite[Section 4]{BD} the authors show graphically a scheme of all the possible $2^6$ Hopf data in $\C$. The analogous scheme can be considered in a general $\K$.

\subsection{When a Hopf datum is a paired wreath}

For the converse of \coref{Hopf data} one should prove that given a Hopf datum, the points 2, 5 and 6 in the definition of a paired wreath hold true.
To begin with, we will analyze which part of a Hopf datum assures that
$(F, \psi: BF\to FB, \mu_M, \eta_M)$ is a left wreath around $B$ and $(B, \phi': FB\to BF, \Delta_C', \Epsilon_C')$ is a right cowreath around $F$,
{\em i.e.} that the point 2 is fulfilled. We will study separate cases.
Before we sum up our findings, we introduce:

\begin{defn} \delabel{mHd}
A {\em monad Hopf datum} in $\K$ consists of the following data:
\begin{enumerate}
\item a monad $(B,
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gob{2}{B}
\gend,
\gbeg{1}{2}
\gu{1} \gnl
\gob{1}{B}
\gend)$, a 1-cell $F$, both over the same 0-cell $\A$ in $\K$, with 2-cells
$\gbeg{2}{3}
\got{2}{B} \gnl
\gcmu \gnl
\gob{1}{B} \gob{1}{B}
\gend,
\gbeg{1}{2}
\got{1}{B} \gnl
\gcu{1} \gnl
\gend$ and
$\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gob{2}{F}
\gend,
\gbeg{1}{2}
\gu{1} \gnl
\gob{1}{F} \gnl
\gend,
\gbeg{2}{3}
\got{2}{F} \gnl
\gcmu \gnl
\gob{1}{F} \gob{1}{F}
\gend,
\gbeg{1}{2}
\got{1}{F} \gnl
\gcu{1} \gnl
\gend$ in $\K$, satisfying the compatibility conditions:
$
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,22cm} \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{B} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend, \hspace{0,5cm}
\gbeg{1}{3}
 \gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gnl
\gend, \hspace{0,2cm}
\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{2}{} \gnl
\gend B=
\Id_{id_{\A}},
\hspace{0,7cm}
\gbeg{1}{3}
 \gu{1} \gnl
\hspace{-0,34cm} \gcmu \gnl
\gob{1}{F} \gob{1}{F} \gnl
\gend=
\gbeg{2}{3}
\gu{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{F}
\gend, \hspace{0,2cm}
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gmu \gnl
\gvac{1} \hspace{-0,22cm} \gcu{1} \gnl
\gend=
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend, \hspace{0,2cm}
\gbeg{1}{2}
\gu{1} \gnl
\gcu{1} \gnl
\gob{2}{} \gnl
\gend F=
\Id_{id_{\A}};
$$
\item further 2-cells:
\begin{equation} \eqlabel{module cells}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F}
\gend, \hspace{0,6cm}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gob{1}{B}
\gend
\end{equation}
so that
$\gbeg{2}{1}
\glm \gnl
\gend$ makes $F$ a proper left $B$-module, and the following relations are fulfilled:
$$
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gvac{1} \gcu{1}
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend,
\hspace{0,8cm}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gcu{1}
\gend=
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\gcl{1} \gcl{1} \gnl
\gcu{1} \gcu{1} \gnl
\gend
$$
\item 2-cells:
$$
\gbeg{2}{3}
\got{1}{F} \got{1}{F} \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gob{3}{B}
\gend
$$
and
$$\tau_{B,F}:BF\to FB, \tau_{F,B}:FB\to BF, \tau_{B,B}:BB\to BB, \tau_{F,F}:FF\to FF$$
where $\tau_{B,F}, \tau_{F,F}, \tau_{F,F}, \tau_{F,B}$ are left and right monadic and comonadic distributive laws, where the adjectives ``monadic'' and ``monadic''
are meant with respect to the 2-cells from the point 1.;
\item so that setting:
$$
\psi=
\gbeg{4}{5}
\got{2}{B} \got{2}{F} \gnl
\gcmu \gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\glm \grmo \gcl{1} \gnl
\gob{1}{} \gob{1}{F} \gob{1}{B}
\gend,
\hspace{0,7cm}
\mu_M=
\gbeg{5}{5}
\got{2}{F} \got{2}{F} \gnl
\gcmu \gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gob{2}{F} \gob{3}{B}
\gend
$$
the identities \equref{mod alg} -- \equref{mod alg unity}, \equref{weak assoc. mu_M}--\equref{weak unity eta_M}, \equref{F mod alg} -- \equref{F mod alg unit},
\equref{weak action} -- \equref{weak action unity}, \equref{2-cocycle condition} -- \equref{normalized 2-cocycle} and \equref{epsilon to sigma} hold true.
\end{enumerate}
\end{defn}

\begin{rem}
In the above Definition we could put
$$
\mu_M=
\gbeg{5}{7}
\got{1}{} \got{1}{F} \got{4}{F} \gnl
\gwcm{3} \gcmu \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \gcl{3} \gnl
\gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{1} \glm \gcl{1} \gnl
\gwmu{3} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gob{1}{F} \gob{5}{B}
\gend
$$
assuming the existence of the 2-cell
$\gbeg{2}{1}
\grmo \gvac{1} \gcl{1}
\gend$ and the relation
$
\gbeg{2}{4}
\got{3}{F} \gnl
\grmo \gvac{1} \gcl{1}\gnl
\gcl{1} \gcu{1} \gnl
\gob{1}{B}
\gend=
\gbeg{1}{4}
\got{1}{F} \gnl
\gcu{1} \gnl
\gu{1} \gnl
\gob{1}{B}
\gend
$. While on one hand, these come from the comonadic structures, on the other hand, the cases for which we will prove to be wreaths it will happen that either $\sigma$ is trivial
(thus $\mu_M$ is canonical), or that
$\gbeg{2}{1}
\glm \gnl
\gend$ is trivial. In both cases we will have the form of $\mu_M$ as in the above Definition.
\end{rem}

\medskip

Similarly as in \coref{Hd conseq}, we have:

\begin{cor} \colabel{monad Hd conseq}
In a monad Hopf datum the following identity holds true:
$$
\gbeg{2}{4}
\got{3}{F} \gnl
\gu{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\psi} \grmptb \gnl
\gob{1}{F} \gob{1}{B}
\gend=
\gbeg{2}{4}
\got{1}{F} \gnl
\gcl{1} \gu{1} \gnl
\gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{B}
\gend
$$
\end{cor}

A {\em comonad Hopf datum} is defined dually, using ``$\pi$-symmetry''.

\medskip

\begin{rem} \rmlabel{wreath datum}
We will also want to name the direct consequences of a wreath. Assume that $B$ is a monad. The data in point 4 in the above Definition, excluding the identity \equref{epsilon to sigma},
we will call a {\em wreath datum}, assuming that all the appearing 2-cells exist.
\end{rem}

\medskip

We will also need:

\begin{defn}
We say that $\tau_{B,F}$ is natural with respect to
$\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gob{1}{B} \gnl
\gend$,
$\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F} \gnl
\gend$ and $\sigma$, respectively, if it holds:
\begin{center}
\begin{tabular}{p{4.7cm}p{0cm}p{4.7cm}p{0cm}p{4.7cm}}
\begin{equation} \eqlabel{nat rm}
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\grmo \gcl{1} \gvac{1} \gcl{1} \gnl
\glmptb \gnot{\tau_{B,F}} \gcmp \grmptb \gnl
\gcl{1} \gvac{1} \gcl{1} \gnl
\gob{1}{F} \gob{3}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gcl{1} \grmo \gcl{1}  \gnl
\gob{1}{F} \gob{1}{B}
\gend
\end{equation} & &
\begin{equation} \eqlabel{nat lm}
\gbeg{3}{5}
\got{1}{B} \got{1}{B} \got{1}{F} \gnl
\gcl{1} \glm \gnl
\glmptb \gnot{\tau_{B,F}} \gcmp \grmptb \gnl
\gcl{1} \gvac{1} \gcl{1} \gnl
\gob{1}{F} \gob{3}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gnl
\glm \gcl{1} \gnl
\gvac{1} \gob{1}{F} \gob{1}{B}
\gend
\end{equation} & &
\begin{equation} \eqlabel{nat sigma}
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gob{1}{F} \gob{3}{B}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gnl
\gvac{1} \gcl{1} \gcl{1} \gnl
\gvac{1} \gob{1}{F} \gob{1}{B}
\gend
\end{equation}
\end{tabular}
\end{center}
respectively. Substituting $\tau_{B,F}$ in the above three diagrams by $\tau_{B,B}, \tau_{F,F}$ and $\tau_{B,B}$, respectively, we obtain the notions of naturality for the pairs
$(\tau_{B,B},
\gbeg{2}{1}
\grmo \gcl{1} \gnl
\gend),
 (\tau_{F,F},
\gbeg{2}{1}
\glm \gnl
\gend)$ and $(\tau_{B,B}, \sigma)$, respectively, in the obvious way.

We say that $\tau_{F,B}$ is natural with respect to
$\gbeg{2}{3}
\got{3}{F} \gnl
\grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F} \gnl
\gend$,
$\gbeg{2}{3}
\got{1}{B} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F} \gnl
\gend$ and $\rho'$, respectively, if it holds:
\begin{center}
\begin{tabular}{p{4.7cm}p{0cm}p{4.7cm}p{0cm}p{4.7cm}}
\begin{equation} \eqlabel{nat lcm}
\gbeg{3}{5}
\got{1}{F} \got{3}{B} \gnl
\gcl{1} \gvac{1} \gcl{1} \gnl
\glmptb \gnot{\tau_{F,B}} \gcmp \grmptb \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{F}
\gend=
\gbeg{3}{5}
\got{1}{} \got{1}{F} \got{1}{B} \gnl
\grmo \gvac{1} \gcl{1} \gcl{1}\gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{F}
\gend
\end{equation} & &
\begin{equation} \eqlabel{nat rcm}
\gbeg{3}{5}
\got{1}{F} \got{3}{B} \gnl
\gcl{1} \gvac{1} \gcl{1} \gnl
\glmptb \gnot{\tau_{F,B}} \gcmp \grmptb \gnl
\grcm \gcl{1} \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{F}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \gnl
\gcl{1} \grcm \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{F}
\gend
\end{equation} & &
\begin{equation} \eqlabel{nat rho'}
\gbeg{3}{5}
\got{1}{F} \got{3}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\rho'} \grmpb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{B}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \gnl
\gcl{1} \gcl{1} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F, B}} \grmptb \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\rho'} \grmpb \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{B}
\gend
\end{equation}
\end{tabular}
\end{center}
respectively. Substituting $\tau_{F,B}$ in the above three diagrams by $\tau_{F,F}, \tau_{B,B}$ and $\tau_{F,F}$, respectively, we obtain the notions of naturality for the pairs
$(\tau_{F,F},
\gbeg{2}{1}
\grmo \gvac{1} \gcl{1}  \gnl
\gend),
 (\tau_{B,B},
\gbeg{2}{1}
\grcm \gnl
\gend)$ and $(\tau_{F,F}, \rho')$, respectively, in the obvious way.
\end{defn}

\begin{defn}
We say that $(\tau_{F,B}, \tau_{F,F}, \tau_{F,B}), (\tau_{B,F}, \tau_{B,B}, \tau_{F,B}),
(\tau_{B,F}, \tau_{F,F}), (\tau_{F,B}, \tau_{B,B}), \tau_{B,B}$ and $\tau_{F,F}$ satisfy the Yang-Baxter equations if
\begin{center}
\begin{tabular}{p{6cm}p{0cm}p{6cm}}
\begin{equation} \eqlabel{YBE FBF}
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \got{1}{F} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gnl
\gob{1}{F} \gob{1}{B} \gob{1}{F}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gob{1}{F} \gob{1}{B} \gob{1}{F}
\gend
\end{equation} & &
\begin{equation} \eqlabel{YBE BFB}
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{B} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gnl
\gob{1}{B} \gob{1}{F} \gob{1}{B}
\gend
\end{equation}
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{p{6cm}p{0cm}p{6cm}}
\begin{equation} \eqlabel{YBE1}
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gnl
\gob{1}{F} \gob{1}{F} \gob{1}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gob{1}{F} \gob{1}{F} \gob{1}{B}
\gend
\end{equation} & &
\begin{equation} \eqlabel{YBE2}
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{F}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \got{1}{B} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{p{6cm}p{0cm}p{6cm}}
\begin{equation} \eqlabel{YBE BB}
\gbeg{3}{5}
\got{1}{B} \got{1}{B} \got{1}{B} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{B} \got{1}{B} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gob{1}{B} \gob{1}{B} \gob{1}{B}
\gend
\end{equation} & &
\begin{equation} \eqlabel{YBE FF}
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gob{1}{F} \gob{1}{F} \gob{1}{F}
\gend=
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gob{1}{F} \gob{1}{F} \gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center}
hold, respectively.
\end{defn}

Observe that assertions for $\phi'$ and $\rho'$ (and their proofs) are $\pi$-symmetric to those for $\psi$ and $\sigma$. We find:

\begin{prop}  \prlabel{simplest case}
The simplest monad Hopf datum, given by $\psi=\tau_{B,F}$ and trivial $\sigma$ (the actions \equref{module cells} are trivial),
consists of monads $B$ and $F$. It determines a canonical wreath.
\end{prop}

\begin{proof}
In this case, all the identities in the wreath datum, but the two ones claiming the monad laws for $F$, hold trivially. The proof of the rest is direct:
every pair of monads delivers a canonical wreath.
\qed\end{proof}

As we saw, wreaths define associative wreath products whose product $\nabla_{FB}$ and unit $\eta_{FB}$ are given via \equref{wreath (co)product - old}.
It is directly proved that when $\mu_M$ is canonical and $\nabla_{FB}$ and $\eta_{FB}$ define a monad, then $\psi$ is a left and right monadic distributive law.
Then for $\K=\hat\C$, where $\C$ is a monoidal category
we recover the following well-known fact:

\begin{cor}
Given algebras $B,F$ and a morphism $\tau_{B,F}: B\ot F\to F\ot B$ in a monoidal category $\C$,
the tensor product $F\ot B$ has an asssociative product defined by $(\nabla_B\ot\nabla_F)(B\ot\tau_{B,F}\ot F)$, where
$\nabla_B, \nabla_F$ are multiplications of $B,F$, respectively, if and only if $\tau_{B,F}$ is a left and right monadic distributive law in $\C$.
\end{cor}

\begin{prop} \prlabel{triv psi, nontriv sigma}
The monad Hopf datum given by $\psi=\tau_{B,F}$ and a non-trivial $\sigma$ consists of the following 2-cocycle:
$$
\gbeg{3}{6}
\got{1}{B} \got{1}{F} \got{1}{F} \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\sigma} \grmpt \gcl{1} \gnl
\gwmu{3} \gnl
\gob{3}{B}
\gend=
\gbeg{3}{5}
\got{1}{B} \got{1}{F} \got{1}{F}\gnl
\gcl{1} \gcl{1} \gcl{1} \gnl
\gcl{1} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb  \gnl
\gwmu{3} \gnl
\gob{3}{B}
\gend,
\hspace{0,4cm}
\gbeg{5}{7}
\got{1}{F} \got{2}{F} \got{2}{F}\gnl
\gcl{1} \gcmu \gcmu  \gnl
\gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau} \grmptb \gcl{1} \gnl
\gcn{1}{1}{1}{2} \gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb  \gnl
\gvac{1} \hspace{-0,22cm} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gvac{1} \gcn{1}{1}{2}{1} \gnl
\gvac{2} \gwmu{3} \gnl
\gob{7}{B}
\gend=
\gbeg{6}{8}
\got{2}{F} \got{2}{F} \got{2}{F} \gnl
\gcmu \gcmu \gcn{1}{2}{2}{2} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau} \grmptb \gcl{1} \gnl
\gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcn{1}{1}{2}{1} \gnl
\gcn{1}{1}{2}{3} \gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau} \grmptb \gnl
\gvac{1} \glmpt \gnot{\sigma} \gcmpb \grmpt \gcl{1} \gnl
\gvac{2} \gwmu{3} \gnl
\gob{7}{B}
\gend,
\hspace{0,4cm}
\gbeg{2}{5}
\got{3}{F}\gnl
\gu{1} \gcl{1} \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gcl{1} \gnl
\gob{3}{B}
\gend=
\gbeg{1}{4}
\got{1}{F}\gnl
\gcu{1} \gnl
\gu{1} \gnl
\gob{1}{B}
\gend=
\gbeg{3}{5}
\got{1}{F}\gnl
\gcl{1} \gu{1} \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gcl{1} \gnl
\gob{3}{B}
\gend,
\hspace{0,4cm}
\gbeg{2}{3}
\got{1}{F} \got{1}{F}\gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{1} \gcu{1} \gnl
\gend=
\gbeg{2}{2}
\got{1}{F} \got{1}{F}\gnl
\gcu{1} \gcu{1} \gnl
\gend.
$$
If moreover $F$ is a comonad and \equref{YBE1}, \equref{YBE FF} and \equref{nat sigma} hold, then this monad Hopf datum is a wreath.
\end{prop}

\begin{proof}
The first statement is clear the rest of the identities in a monad Hopf datum are trivial in this setting. The second assertion is a particular case of the
analogous claim for $\psi_2$ in \prref{psi_1,2}, which will be proved below.
\qed\end{proof}

\begin{ex} \exlabel{group coh}
When $\K=\hat\C$ where $\C$ is the monoidal (sub)category of trivial modules $M=B$ over a group $G=F$, the above recovers 2-cocycles from the group cohomology. The 2-cell $\tau$ is the flip
in the category of abelian groups and the first condition above reflects the abelianity of the module $M=B$, as an additive group. The second one is the cocycle condition, and the
third one states that $\sigma$ is normalized.
\end{ex}

\begin{prop} \prlabel{psi_1,2}
\begin{enumerate}
\item Set $\psi_1=
\gbeg{3}{5}
\got{2}{B} \got{1}{F} \gnl
\gcmu \gcl{1} \gnl
\gcl{1}  \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gnl
\glm \gcl{1} \gnl
\gob{1}{} \gob{1}{F} \gob{1}{B}
\gend$ (that is,
$\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gob{1}{B}
\gend$ is trivial). If $(B,F)$ is a monad Hopf datum, $\sigma$ is trivial and \equref{proj B-bialg} and \equref{nat lm} hold,
then $(F, \psi_1, \mu_M \hspace{0,2cm}\textnormal{canonical})$ is a left wreath around $B$.

\item Set $\psi_2=
\gbeg{3}{5}
\got{1}{B} \got{2}{F} \gnl
\gcl{1} \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gcl{1} \grmo \gcl{1} \gnl
\gob{1}{F} \gob{1}{B}
\gend$ (that is, $\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F}
\gend$ is trivial). If $(B,F)$ is a monad Hopf datum, $F$ is a comonad, \equref{proj F-bialg}, \equref{YBE1}, \equref{nat rm}, \equref{YBE FF} and \equref{nat sigma} hold,
then $(F, \psi_2, \mu_M)$ is a left wreath around $B$.

\item Set $\phi_1'=
\gbeg{3}{5}
\got{1}{F} \got{1}{B} \gnl
\gcl{1} \grcm \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{B} \gob{2}{F}
\gend$ (that is, $\gbeg{2}{3}
\got{3}{F} \gnl
\grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend$ is trivial). If $(B,F)$ is a comonad Hopf datum, $\rho'$ is trivial and \equref{proj F-bialg} and \equref{nat rcm} hold,
then $(B, \phi'_1, \Delta_C' \hspace{0,2cm}\textnormal{canonical})$ is a right cowreath around $F$.

\item Set $\phi'_2=
\gbeg{3}{5}
\got{1}{} \got{1}{F} \got{1}{B} \gnl
\grmo \gvac{1} \gcl{1} \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gnl
\gmu \gcl{1} \gnl
\gob{2}{B} \gob{1}{F}
\gend$ (that is, $\gbeg{2}{3}
\got{1}{B} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F}
\gend$ is trivial). If $(B,F)$ is a comonad Hopf datum, $B$ is a monad, \equref{proj B-bialg}, \equref{YBE2}, \equref{nat lcm}, \equref{YBE BB} and \equref{nat rho'} hold,
then $(B, \phi'_2, \Delta_C')$ is a right cowreath around $F$.

\end{enumerate}
\end{prop}

\begin{proof}
The parts 3 and 4 are $\pi$-symmetric to 1 and 2, respectively. The statement for $\psi_1$ is straightforward to prove, as well as that $\psi_2$ satisfies the $\psi$ axioms. We show only that
\equref{proj F-bialg}, \equref{weak action}--\equref{weak action unity}, \equref{YBE1}, \equref{nat rm} imply \equref{2-cell mu_M}. The proof that the identities
\equref{2-cocycle condition}, \equref{weak assoc. mu_M}, \equref{YBE FF}, \equref{nat sigma} imply \equref{monad law mu_M} is similar. The necessary identities
for the (co)unities are proved straightforwardly. (Observe that in this setting the identities \equref{mod alg}--\equref{mod alg unity} are trivial and that
\equref{weak assoc. mu_M}--\equref{weak unity eta_M} mean that $F$ is a monad. Consequently, in this setting $F$ is a left bimonad in $\K$.)

We start with the identity \equref{weak action}, compose it from the left with $FF$, then from above with \equref{precompose} and from below with $\gbeg{3}{3}
\got{1}{F} \got{1}{F} \got{1}{B} \gnl
\gmu \gcl{1} \gnl
\gob{2}{F} \gob{1}{B} \gnl
\gend$. By coassociativity of $F$ and distributive law of $\tau_{F,F}$ with respect to the comultiplication of $F$ this yields \equref{L=R 1}: \vspace{1,8cm}
\begin{center} \hspace{-1,3cm}
\begin{tabular}{p{6cm}p{0cm}p{9cm}}
\begin{equation} \eqlabel{precompose}
\gbeg{3}{5}
\got{1}{B} \got{2}{F} \got{2}{F} \gnl
\gcl{1} \gcmu \gcmu \gnl
\glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gcl{1} \gnl
\gob{1}{F} \gob{1}{F} \gob{1}{B} \gob{1}{F} \gob{1}{F} \gnl
\gend
\end{equation} & & \vspace{-1,8cm}
\begin{equation} \eqlabel{L=R 1}
\gbeg{7}{13}
\got{1}{B} \got{4}{F} \got{2}{F} \gnl
\gcl{2} \gvac{1} \hspace{-0,34cm} \gwcm{3} \gvac{1} \hspace{-0,34cm} \gcmu \gnl
\gvac{2} \gcmu \hspace{-0,2cm} \gvac{1} \gcl{1} \gcmu \gcn{1}{6}{0}{0} \gnl
\gvac{2} \hspace{-0,3cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{4} \gnl
\gvac{3} \hspace{-0,3cm} \gcl{1} \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gvac{3} \gcn{2}{1}{2}{3} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gcl{1} \gnl
\gvac{4} \gmu \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{4} \gcn{2}{5}{2}{2} \gcl{1} \grmo \gcl{1} \gcn{1}{1}{3}{1} \gnl
\gvac{6} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{2}{1} \gnl
\gvac{6} \gcl{1} \gcl{1} \grmo \gcl{1} \gnl
\gvac{6} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gcl{1} \gnl
\gvac{7} \gmu \gnl
\gob{10}{B} \gob{-4}{F}
\gend=
\gbeg{7}{10}
\got{1}{B} \got{4}{F} \got{2}{F} \gnl
\gcl{2} \gvac{1} \hspace{-0,34cm} \gwcm{3} \gvac{1} \hspace{-0,34cm} \gcmu \gnl
\gvac{2} \gcmu \hspace{-0,2cm} \gvac{1} \gcl{1} \gcmu \gcn{1}{1}{0}{0} \gnl
\gvac{2} \hspace{-0,3cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gcn{1}{1}{0}{1} \gnl
\gvac{3} \hspace{-0,3cm} \gcl{1} \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1}  \gnl
\gvac{3} \gcn{2}{1}{2}{3} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{4} \gmu \gcl{1} \gcn{1}{1}{2}{1} \gvac{2} \gcl{2} \gnl
\gvac{5} \gcn{1}{2}{0}{0} \grmo \gcl{1} \gnl
\gvac{6} \gwmu{5} \gnl
\gvac{4} \gob{2}{B} \gvac{2} \gob{1}{F}
\gend
\end{equation}
\end{tabular}
\end{center}
Now by the Yang-Baxter equation \equref{YBE1} the left side of \equref{L=R 1} equals:
$$
\gbeg{7}{11}
\got{1}{B} \got{3}{F} \got{3}{F} \gnl
\gcl{2} \gvac{1} \hspace{-0,44cm} \gwcm{3} \gvac{1} \hspace{-0,34cm} \gcmu \gnl
\gvac{2} \gcmu \hspace{-0,2cm} \gvac{1} \gcl{1} \gcmu \gcn{1}{5}{0}{0} \gnl
\gvac{2} \hspace{-0,3cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{3} \gnl
\gvac{3} \hspace{-0,3cm} \gcl{3} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{2}{1} \gcn{1}{1}{2}{1} \gnl
\gvac{4} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{4} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \grmo \gcl{1} \gcn{1}{1}{4}{1} \gnl
\gvac{3} \gmu \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{3}{1} \gnl
\gvac{4} \gcn{1}{2}{0}{0} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb  \grmo \gcl{1} \gnl
\gvac{6} \gmu \gnl
\gob{8}{B} \gob{-2}{F}
\gend\stackrel{\tau}{=}
\gbeg{7}{11}
\gvac{1} \got{1}{B} \got{2}{F} \got{4}{F} \gnl
\gvac{1} \gcl{1} \gcmu \gvac{1}  \gcmu \gnl
\gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \hspace{-0,2cm} \gvac{1} \gcn{1}{1}{0}{1} \gcmu \gcn{1}{5}{0}{0} \gnl
\gvac{1} \gcmu \gcn{1}{1}{0}{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{3} \gnl
\gvac{1} \gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \grmo \gcl{1} \gnl
\gvac{1} \gmu \gcl{2} \gcl{1} \gcn{1}{1}{3}{1} \gnl
\gvac{1} \gcn{1}{1}{2}{2} \gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{2}{1} \gnl
\gvac{2} \gcn{1}{2}{0}{0} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb  \grmo \gcl{1} \gnl
\gvac{4} \gmu \gnl
\gob{4}{B} \gob{2}{F}
\gend\stackrel{\equref{nat rm}}{=}
\gbeg{7}{9}
\gvac{1} \got{1}{B} \got{2}{F} \got{3}{F} \gnl
\gvac{1} \gcl{1} \gcmu \gvac{1} \hspace{-0,22cm} \gcmu \gnl
\gvac{2} \hspace{-0,22cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gcmu \gcn{1}{1}{0}{1} \gnl
\gvac{2} \gcn{1}{1}{1}{0} \grmo \gcl{1} \gcn{1}{1}{3}{1} \gcn{1}{2}{3}{1} \gvac{1} \gcl{1} \gnl
\gvac{1} \gcmu \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gvac{1}  \gcn{1}{2}{3}{1} \gnl
\gvac{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb  \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb  \gnl
\gvac{1} \gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb  \grmo \gcl{1} \gnl
\gvac{1} \gcn{1}{1}{2}{2} \gvac{2} \gmu \gnl
\gob{4}{B} \gob{2}{F}
\gend\stackrel{\tau}{=}
\gbeg{7}{12}
\gvac{2} \got{1}{B} \got{2}{F} \got{1}{F} \gnl
\gvac{2} \gcl{1} \gcmu \gcl{2} \gnl
\gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb\gcl{1} \gnl
\gvac{2} \gcl{1} \grmo \gcl{1} \gcn{1}{1}{3}{2} \gnl
\gvac{2} \gcl{1} \gcl{1} \gcmu \gnl
\gvac{2} \gcn{1}{2}{1}{-2}  \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{3} \gcn{1}{1}{1}{0} \grmo \gcl{1} \gnl
\gcmu \gcmu \gcl{3} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gcn{1}{1}{2}{2} \gvac{2} \gmu \gnl
\gob{2}{B}  \gob{4}{F}
\gend
$$
which is the left hand-side of \equref{2-cell mu_M} in the present setting. In the first and the third equation we applied the distributive law of $\tau_{B,F}$ with respect
to the comultiplication of $F$. The right hand-side of \equref{L=R 1}, by the distributivity law of $\tau_{B,F}$ with respect to the multiplication of $F$, equals:
$$
\gbeg{8}{10}
\got{1}{B} \got{4}{F} \got{2}{F} \gnl
\gcl{4} \gvac{1} \hspace{-0,34cm} \gwcm{3} \gvac{1} \hspace{-0,34cm} \gcmu \gnl
\gvac{2} \gcmu \hspace{-0,2cm} \gvac{1} \gcl{1} \gcmu \gcn{1}{1}{0}{0} \gnl
\gvac{3} \gcn{1}{1}{0}{0} \gcn{1}{1}{0}{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gcn{1}{1}{0}{1} \gnl
\gvac{3} \hspace{-0,3cm} \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1}  \gnl
\gvac{2} \gcn{2}{1}{2}{4} \gmu \gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{4} \hspace{-0,22cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{3}{1} \gvac{2} \gcn{1}{1}{2}{2} \gnl
\gvac{4} \gcl{2} \grmo \gcl{1} \gvac{3} \gcn{1}{1}{2}{1} \gnl
\gvac{5} \gwmu{5} \gnl
\gvac{4} \gob{1}{B} \gvac{2} \gob{1}{F}
\gend\stackrel{\tau}{=}
\gbeg{8}{10}
\got{1}{B} \got{4}{F} \got{3}{F} \gnl
\gcl{4} \gvac{1} \hspace{-0,34cm} \gwcm{3} \gvac{1} \gcmu \gnl
\gvac{2} \gcl{1} \gvac{1} \glmptb \gnot{\tau_{F,F}} \gcmp \grmptb \gcl{1} \gnl
\gvac{2} \hspace{-0,3cm} \gcmu \gcmu \gvac{1} \hspace{-0,22cm} \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{3} \gcn{1}{1}{0}{0} \hspace{-0,22cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gvac{1} \gcn{1}{3}{2}{2} \gnl
\gvac{2} \gcn{1}{1}{1}{2} \gmu \gmu \gnl
\gvac{3} \hspace{-0,22cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{3}{1} \gnl
\gvac{3} \gcl{2} \grmo \gcl{1} \gvac{3} \gcn{1}{1}{3}{1} \gnl
\gvac{4} \gwmu{5} \gnl
\gvac{3} \gob{1}{B} \gvac{2} \gob{1}{F}
\gend \stackrel{\equref{proj F-bialg}}{=}
\gbeg{6}{9}
\got{2}{B} \got{1}{F} \got{3}{F} \gnl
\gcn{1}{4}{2}{2} \gvac{1} \hspace{-0,34cm} \gcmu \gcmu \gnl
\gvac{2} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gvac{2} \gmu \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gvac{2} \gcmu \gvac{1} \gcl{3} \gnl
\gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{1} \gcl{2} \grmo \gcl{1} \gnl
\gvac{2} \gwmu{4} \gnl
\gvac{1} \gob{1}{B} \gvac{1} \gob{2}{F}
\gend
$$
and this is precisely the right hand-side of \equref{2-cell mu_M}. In the first equation we applied the distributivity law of $\tau_{F,F}$ with respect to the comultiplication of $F$.
\qed\end{proof}

The statement in the above Proposition in the setting of $\psi_2$ can be reformulated as follows:

\begin{prop} \prlabel{Sw iff}
Suppose that $(B,F, \psi_2, \mu_M)$ is a wreath system (equivalently, that $F$ is a monad and that
$B$ is a right $F$-module monad, $\sigma$ is a normalized 2-cocycle and the $F$-action on $B$ is twisted by $\sigma$) and that $F$ is a left bimonad.
Then $(B,F, \psi_2, \mu_M)$ is a wreath.

Consequently, $(B,F, \psi_2, \mu_M)$ is a wreath if and only if it is a wreath system and $F$ is a left bimonad in $\K$.
\end{prop}

\bigskip

(In the following Proposition we will label the 2-cell $\psi$ as $\psi_4$, as the notation $\psi_3$ we reserved for a 2-cell $\psi$ appearing in a mixed biwreath-like object in \cite{Femic5}.)

\begin{prop} \prlabel{psi_4}
Set $\psi_4=
\gbeg{4}{5}
\got{2}{B} \got{2}{F} \gnl
\gcmu \gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\glm \grmo \gcl{1} \gnl
\gob{1}{} \gob{1}{F} \gob{1}{B}
\gend$
and suppose we are in a setting where the following assumptions hold true:
\begin{enumerate}
\item $(B,F)$ is a monad Hopf datum and $F$ is a comonad;
\item there are 2-cells
$
\gbeg{2}{3}
\got{3}{F} \gnl
\grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F}
\gend,
\hspace{0,3cm}
\gbeg{2}{3}
\got{1}{B} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F}
\gend,
\hspace{0,3cm}
\gbeg{3}{3}
\got{3}{F} \gnl
\glmpb \gnot{\hspace{-0,34cm}\rho'} \grmptb \gnl
\gob{1}{B}\gob{1}{B}
\gend
$ such that \equref{proj F-bialg}, \equref{proj B-bialg} and the second and third equation in \equref{1-3} hold;
\item $B$ is a comonad (or, \equref{weak coass. Delta_C'}--\equref{weak counit Epsilon_C'} hold and
$\gbeg{2}{3}
\got{1}{B} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F} \gnl
\gend$ is trivial or $\rho'$ is trivial (hence $\Delta_C'$ is canonical));
\item $\sigma$ is trivial (hence $\mu_M'$ is canonical);
\item $\tau_{B,F}$ is natural with respect to
$\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gob{1}{B} \gnl
\gend$ and
$\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F} \gnl
\gend$ (recall \equref{nat rm}, \equref{nat lm}).
\end{enumerate}
Then $(F, \psi_4, \mu_M \hspace{0,2cm}\textnormal{canonical})$ is a left wreath around $B$.
\end{prop}

\begin{proof}
Let us prove that $\psi_4$ satisfies the $\psi$ axioms. Indeed:
$$\hspace{-0,2cm}
\gbeg{3}{5}
\got{1}{B} \got{1}{B} \got{1}{F} \gnl
\gmu \gcn{1}{1}{1}{0} \gnl
\gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\widetilde{\psi}} \grmptb \gnl
\gvac{1} \gcl{1} \gcl{1} \gnl
\gvac{1} \gob{1}{F} \gob{1}{B}
\gend=
\scalebox{0.84}[0.84]{
\gbeg{5}{6}
\got{1}{B} \got{1}{B} \got{2}{F}\gnl
\gmu \gcn{1}{1}{2}{2} \gnl
\gcmu \gcmu \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\glm \grmo \gcl{1} \gnl
\gvac{1} \gob{1}{F} \gob{1}{B}
\gend}
\stackrel{\equref{proj B-bialg}}{=}
\scalebox{0.84}[0.84]{
\gbeg{9}{9}
\gvac{1} \got{1}{B} \gvac{2} \got{1}{B} \got{3}{F}\gnl
\gwcm{3} \gwcm{3} \gcl{3} \gnl
\gcl{1} \gvac{1} \gcl{1} \grcm \gcl{3} \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1}\gnl
\gcl{1} \gvac{1} \gcl{1} \grmo \gcl{1} \gvac{1} \gcn{1}{1}{3}{2} \gnl
\gwmu{3} \gwmu{3} \hspace{-0,4cm} \gcmu \gnl
\gvac{2} \gcn{1}{1}{1}{5} \gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{4} \glm \grmo \gcl{1} \gnl
\gvac{5} \gob{1}{F} \gob{1}{B}
\gend}\stackrel{\tau}{=}
\scalebox{0.84}[0.84]{
\gbeg{9}{10}
\gvac{1} \got{1}{B} \gvac{2} \got{1}{B} \got{3}{F}\gnl
\gwcm{3} \gwcm{3} \gcl{3} \gnl
\gcl{1} \gvac{1} \gcl{1} \grcm \gcl{2} \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gnl
\gcl{1} \gvac{1} \gcl{1} \grmo \gcl{1} \gvac{1}  \gcn{1}{1}{1}{0} \hspace{-0,22cm} \gcmu  \gnl
\gvac{1} \hspace{-0,2cm} \gwmu{3} \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{2} \gnl
\gvac{2} \gcn{1}{1}{2}{5} \gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{4} \glm \gmu \gcn{1}{1}{1}{0} \gnl
\gvac{5} \gcl{1} \gvac{1} \hspace{-0,34cm} \grmo \gcl{1} \gnl
\gvac{5} \gob{2}{F} \gob{1}{B}
\gend}
$$

$$\stackrel{\equref{F mod alg}}{=}
\scalebox{0.84}[0.84]{
\gbeg{10}{13}
\gvac{1} \got{1}{B} \gvac{2} \got{1}{B} \got{3}{F}\gnl
\gwcm{3} \gwcm{3} \gcl{2} \gnl
\gcl{1} \gvac{1} \gcl{1} \grcm \gcl{2} \gnl
\gcl{1} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gvac{1} \gcn{1}{1}{1}{2}  \gnl
\gcl{1} \gvac{1} \gcl{1} \grmo \gcl{1} \gvac{1} \gcn{1}{1}{1}{0} \hspace{-0,22cm} \gwcm{3}  \gnl
\gvac{1} \hspace{-0,3cm} \gwmu{3} \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gvac{1} \gcl{1} \gnl
\gvac{2} \gcn{1}{1}{2}{5} \gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{1}{2} \gvac{1} \gcn{1}{1}{1}{2}\gnl
\gvac{4} \glm \gcl{1} \gcmu \gcmu \gnl
\gvac{5} \gcl{4} \gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{6} \gcn{1}{1}{1}{3} \glm \grmo \gcl{1} \gnl
\gvac{7} \grmo \gcl{1} \gvac{1} \gcl{1} \gnl
\gvac{7} \gwmu{3} \gnl
\gvac{5} \gob{1}{F} \gob{5}{B}
\gend}
\stackrel{\tau}{\stackrel{coass. F}{\stackrel{B\x mod.}{=}}}
\scalebox{0.84}[0.84]{
\gbeg{11}{12}
\got{2}{B} \gvac{1} \got{1}{B} \got{7}{F}\gnl
\gcmu \gwcm{3} \gvac{2} \gcl{2} \gnl
\gcl{1} \gcl{1} \grcm \gcl{2} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \gvac{2} \gwcm{3} \gnl
\gcl{3} \gcl{3} \grmo \gcl{1} \gvac{1} \hspace{-0,34cm} \gcmu \gcmu \gcn{1}{3}{2}{2} \gnl
\gvac{2} \gcn{1}{1}{2}{2} \gvac{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{3} \gcn{1}{1}{0}{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gcl{1} \gnl
\gcn{1}{2}{2}{5} \hspace{-0,4cm} \gcn{1}{1}{4}{5} \gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{2}{1} \gnl
\gvac{3} \glm \gcn{1}{1}{1}{3}  \glm \grmo \gcl{1} \gnl
\gvac{3} \glm \gvac{1} \grmo \gcl{1} \gvac{1} \gcl{1} \gnl
\gvac{4} \gcl{1} \gvac{1} \gwmu{3} \gnl
\gvac{4} \gob{1}{F} \gob{5}{B}
\gend}
\stackrel{coass. B}{=}
\scalebox{0.84}[0.84]{
\gbeg{11}{10}
\got{2}{B} \gvac{2} \got{1}{B} \got{6}{F}\gnl
\gcmu \gvac{1} \gwcm{3} \gvac{1} \hspace{-0,34cm} \gwcm{3} \gnl
\gvac{1} \hspace{-0,3cm} \gcl{1} \gcl{1} \gwcm{3} \gcl{1} \gcmu \gcn{1}{2}{2}{2} \gnl
\gvac{1} \gcl{1} \gcl{1} \grcm \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{2}{1} \gnl
\gvac{1} \gcl{2} \gcl{2} \grmo \gcl{1} \gvac{1} \gcn{1}{1}{1}{-1} \glm \grmo \gcl{1} \gnl
\gvac{3} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{5}{1} \gcn{1}{2}{5}{3} \gnl
\gvac{1} \gcn{1}{1}{1}{3} \glm \grmo \gcl{1} \gnl
\gvac{2} \glm \gwmu{4}  \gnl
\gvac{3} \gob{1}{F} \gob{4}{B}
\gend}=
$$

$$
\stackrel{\equref{nat rm}}{\stackrel{\tau}{=}}
\scalebox{0.84}[0.84]{
\gbeg{11}{11}
\got{2}{B} \gvac{2} \got{2}{B} \got{4}{F}\gnl
\gcmu \gvac{1} \gwcm{4} \gcmu \gnl
\gcl{5} \gcl{1} \gwcm{3} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb  \gcl{1} \gnl
\gvac{1} \gcl{1} \grcm \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,32cm} \gcmu \hspace{-0,22cm} \grmo \gcl{1} \gvac{1} \gnl
\gvac{2} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,B}} \grmptb \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,3cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gcn{1}{2}{0}{0} \gnl
\gvac{2} \gcn{1}{1}{2}{2} \gcn{2}{1}{2}{3} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \glm  \gnl
\gvac{3} \gcn{1}{1}{0}{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gcn{1}{2}{3}{-1} \gcn{1}{1}{2}{2}  \gnl
\gvac{2} \gcn{1}{1}{0}{3}  \glm \grmo \gcl{1} \gvac{3} \gcn{1}{2}{0}{-1} \gnl
\gvac{3} \glm \grmo \gcl{1} \gnl
\gvac{4} \gcl{1} \gwmu{4}  \gnl
\gvac{4} \gob{1}{F} \gob{4}{B}
\gend}
\stackrel{\equref{nat lm}}{\stackrel{F\x mod.}{=}}
\scalebox{0.84}[0.84]{
\gbeg{11}{11}
\got{2}{B} \gvac{2} \got{2}{B} \got{4}{F}\gnl
\gcmu \gvac{1} \gwcm{4} \gcmu \gnl
\gcl{4} \gcl{3} \gwcm{3} \gvac{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb  \gcl{1} \gnl
\gvac{2} \grcm \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,32cm} \gcmu \hspace{-0,22cm} \grmo \gcl{1} \gvac{1} \gnl
\gvac{3} \gcl{1} \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,3cm} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gcn{1}{4}{0}{0} \gnl
\gvac{3} \gcn{1}{1}{0}{1} \gcn{1}{1}{0}{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \glm  \gnl
\gvac{1} \gcn{2}{2}{2}{5} \gcn{1}{1}{1}{3} \glm \gwmu{3} \gnl
\gvac{4} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcn{1}{1}{3}{1} \gnl
\gvac{3} \glm \grmo \gcl{1} \gvac{3} \gcn{1}{1}{0}{-1} \gnl
\gvac{4} \gcl{1} \gwmu{4}  \gnl
\gvac{4} \gob{1}{F} \gob{4}{B}
\gend}
\stackrel{\equref{1-3}}{=}
\scalebox{0.84}[0.84]{
\gbeg{6}{9}
\got{1}{B} \got{2}{B} \got{2}{F}\gnl
\gcl{3} \gcmu \gcmu \gnl
\gvac{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{1} \glm \grmo \gcl{1} \gnl
\hspace{-0,34cm} \gcmu \gcmu \gcn{1}{2}{0}{0} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\glm \grmo \gcl{1} \gcn{1}{1}{2}{1} \gnl
\gvac{1} \gcl{1} \gmu \gnl
\gvac{1} \gob{1}{F} \gob{2}{B}
\gend}=
\gbeg{4}{5}
\got{1}{B} \got{1}{B} \got{1}{F} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\widetilde{\psi}} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\widetilde{\psi}} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{B}
\gend
$$
We clarify only the brief notations in the above equalities that possibly are not clear enough:
in the third equation we applied the monad distributive law for $\tau$, in the fifth one the comonad distributive law for $\tau$ and left $B$-module
structure of $F$, in the seventh one the comonad distributive law for $\tau$, in the eighth one the right $F$-module structure on $B$ and in the ninth one the third equation in
\equref{1-3}. Observe that in order for the comultiplication of $B$ to be coassociative
we need the third of the above assumptions.
Moreover, in order for the right $F$-action on $B$ to be proper, by \equref{weak action} it should be either trivial (in which case we are in the setting of $\psi_1$ from \prref{psi_4}),
or the fourth assumption above should be fulfilled.

The proof that $\mu_M$ canonical (assuming that $\sigma$ is trivial) satisfies the 2-cell condition is analogous: take left-right symmetric diagrams and interchange the r\^oles of $B$ and $F$.
Recall that we called this {\em $\alpha$-symmetry}. Then the statement to prove, as well as the proof itself, are $\alpha$-symmetric to those from above. One part of the conditions
necessary to prove the $\psi$ axioms is already auto $\alpha$-symmetric, from those conditions which do not contain their $\alpha$-symmetric counterpart,
we need to add: \equref{proj F-bialg}, \equref{mod alg} and the second equation in \equref{1-3}.

The monad law for $\mu_M$ comes down to associativity of $F$, it is fulfilled by \equref{weak assoc. mu_M}, since $\mu_M$ is canonical. The necessary relations for the units are proved
straighforwardly.
\qed\end{proof}

\bigskip

By $\pi$-symmetry, we have:

\begin{prop} \prlabel{phi_4}
Set $\phi'_4=
\gbeg{4}{5}
\got{1}{} \got{1}{F} \got{1}{B} \gnl
\grmo \gvac{1} \gcl{1} \grcm \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,B}} \grmptb \gcl{1} \gnl
\gmu \gmu \gnl
\gob{2}{B} \gob{2}{F}
\gend$. Then $(B, \phi'_4, \Delta_C' \hspace{0,2cm}\textnormal{canonical})$ is a right cowreath around $F$ if the following assumptions hold true:
\begin{enumerate}
\item $(B,F)$ is a comonad Hopf datum and $B$ is a monad;
\item there are 2-cells
$
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F}
\gend,
\hspace{0,3cm}
\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\grmo \gcl{1} \gnl
\gob{1}{B}
\gend,
\hspace{0,3cm}
\gbeg{3}{3}
\got{1}{F} \got{1}{F}  \gnl
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gob{3}{B}
\gend
$ such that \equref{proj F-bialg}, \equref{proj B-bialg} and the first and third equation in \equref{4-6} hold;
\item $F$ is a monad (or, \equref{weak assoc. mu_M}--\equref{weak unity eta_M} hold and
$\gbeg{2}{3}
\got{1}{B} \got{1}{F} \gnl
\glm \gnl
\gob{3}{F} \gnl
\gend$ is trivial or $\sigma$ is trivial (hence $\mu_M$ is canonical);
\item $\rho'$ is trivial (hence $\Delta_C'$ is canonical);
\item $\tau_{F,B}$ is natural with respect to
$\gbeg{2}{3}
\got{3}{F} \gnl
\grmo \gvac{1} \gcl{1} \gnl
\gob{1}{B} \gob{1}{F} \gnl
\gend$ and
$\gbeg{2}{3}
\got{1}{B} \gnl
\grcm \gnl
\gob{1}{B} \gob{1}{F} \gnl
\gend$ (recall \equref{nat lcm}, \equref{nat rcm}).
\end{enumerate}
\end{prop}

\bigskip

To the 2-cells
$\gbeg{2}{1}
\glm \gnl
\gend,
\gbeg{2}{1}
\grmo \gcl{1} \gnl
\gend,
\gbeg{2}{1}
\glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb \gnl
\gend
$ \hspace{0,1cm}
in a monad Hopf datum we will assign numbers 0 or 1, depending on whether the action and cocycle are trivial or not. The first two 2-cells determine $\psi$.
Then a monad Hopf datum determined by the former three 2-cells we will denote shortly by $((i,j),k)$ with $i,j,k\in\{0,1\}$, where the first two entries correspond to the action 2-cells,
and the third one to the cocycle. Similarly, we will assign a 3-tuple $((\crta i,\crta j),\crta k)$, with $\crta i,\crta j,\crta k\in\{0,1\}$,  to the comonad Hopf datum determined by the
2-cells
$\gbeg{2}{1}
\grcm \gnl
\gend,
\gbeg{2}{1}
\grmo \gvac{1} \gcl{1} \gnl
\gend,
\gbeg{3}{1}
\glmpb \gnot{\hspace{-0,34cm}\rho'} \grmptb \gnl
\gend$ in this same order. Neglecting the two simplest (co)monad Hopf data studied above, we present in the following table the rest of monad Hopf data which we studied in \prref{psi_1,2} and
\prref{psi_4} and which, under respective hypotheses, determine a wreath, together with their $\pi$-symmetric companions:
\begin{table}[h!]
\begin{center}
\begin{tabular}{ c c }

 $(\psi, \sigma)$ & \hspace{0,2cm} ($\pi$-symmetric $\phi'$, $\rho'$) \\ [0.5ex]
\hline
$ ((1,0),0)$ & $((\crta 0,\crta 1),\crta 0)$ \\ [1ex]
 $((0,1),0)$ & $((\crta 1,\crta 0),\crta 0)$  \\ [1ex]
 $((0,1),1)$ & $((\crta 1,\crta 0),\crta 1)$  \\ [1ex]
 $((1,1),0)$ & $((\crta 1,\crta 1),\crta 0)$  \\ [1ex]
\end{tabular}
\caption{(Co)monad Hopf data which are (co)wreaths}
\label{table:1}
\end{center}
\end{table}\\
\indent If we are to decide which Hopf data determine a paired wreath, we observe that only $\psi_4$ presented restrictions on its $\pi$-symmetric companion (see condition 3. in \prref{psi_4}).
However, from the above table it is clear that $\psi_4$ may be combined will all the resting four $\phi'$'s. So, from Table \ref{table:1} we possibly have $4\times 4=16$ paired wreaths.
It remains to study when a Hopf datum implies conditions 5 and 6 of a paired wreath.

Let us analyze the condition 5. From \equref{lambda B} we know the form that 2-cell $\lambda_B$ should have. We only may claim, which is directly checked,
that \equref{proj B} is fulfilled if either $\sigma$ is trivial, or $\sigma$ is non-trivial (then we are in the setting of $((0,1),1)$) and
$\gbeg{2}{1}
\grcm \gnl
\gend$ is trivial.
The latter case by $\pi$-symmetry yields that $((\crta 1,\crta 0),\crta 1)$ can coexist in a paired wreath only with
$((0,1),0)$ and $((0,1),1)$. Summing up, from the above 16 combinations the
ones in Table \ref{table:2} certainly fulfill condition 5.
\begin{table}[h!]
\begin{center}
\begin{tabular}{ c c }

 $(\psi, \sigma)$ & \hspace{0,2cm} ((matching cowreaths) \\ [0,5ex]
\hline
 $ ((0,1),1)$ & $((\crta 1,\crta 0),\crta 0), ((\crta 1,\crta 0),\crta 1)$  \\ [1ex]
  $((0,1),0)$ & all 4  \\ [1ex]
$((1,0),0)$ & \hspace{0,8cm}  $((\crta 0,\crta 1),\crta 0), ((\crta 1,\crta 0),\crta 0), ((\crta 1,\crta 1),\crta 0)$  \\ [1ex]
 $((1,1),0)$ & \hspace{0,8cm}  $((\crta 0,\crta 1),\crta 0), ((\crta 1,\crta 0),\crta 0), ((\crta 1,\crta 1),\crta 0)$  \\ [1ex]
\end{tabular}
\caption{(Co)wreaths satisfying \equref{proj B} / \equref{proj F}}
\label{table:2}
\end{center}
\end{table}

As far as for condition 6, consider the case where the four (co)action 2-cells \equref{(co)module cells} are non-trivial and $\sigma$ and $\rho'$ are trivial.
The proof in string diagrams of the fourth relation in \deref{tau-bim} in this setting, with
$\tau_{FB,FB}$ given as in \equref{tau FB}, is pretty tedious and possibly leads to a never ending loop. We proved, however, that if one of the
four (co)action 2-cells is trivial, then the Hopf datum is a bimonad. (Observe that if one proves the result choosing any of the four 2-cells to be trivial, then
the result holds true if any other of the four 2-cells is trivial, because of the auto $\alpha$- and $\pi$-symmetry of a Hopf datum, \rmref{when cocycles are trivial}.)
Instead of presenting here our proof in string diagrams, which is tedious, we make the following observation.
In \cite[Section 2]{BD1} the authors use a remarkable tool in the context of a Hopf datum in a braided monoidal category $\C$ that can fully be taken over and used in our context of
a Hopf datum in a general 2-category $\K$. It serves to study a Hopf datum of the form $((1,1),0)$ with $((\crta 1,\crta 1),\crta 0)$ and it helps to decide when the
identity \equref{paired biwreath biproduct} in this setting is fulfilled. Namely, they introduce a {\em recursive Hopf datum}, they prove in \cite[Theorem 2.14]{BD1}
that for a recursive Hopf datum $(B,F)$ the object $B\ot F$ is a bialgebra in $\C$ (the 1-cell $FB$ is a bimonad in $\K$) and in \cite[Definition and Proposition 2.15]{BD1} they prove that
every {\em trivalent} Hopf datum is a recursive Hopf datum. A trivalent Hopf datum is a one for which $\sigma$ and $\rho'$ are trivial and at most three of the four (co)action 2-cells
\equref{(co)module cells} are non-trivial.
Thus we may state:

\begin{thm} \thlabel{sufficient for bimonad}
Let $(B,F)$ be a Hopf datum in $\K$ determined by $((i,j),0)$ with $((\crta k,\crta l),\crta 0)$ with $i,j,k,l\in\{0,1\}$ and so that $i+j+k+l\leq 3$.
Suppose that the 2-cells $\tau$ satisfy the naturality conditions and Yang-Baxter equations \equref{nat rm} -- \equref{nat sigma}, \equref{YBE FBF} -- \equref{YBE FF}.
Then $FB$ is a $\tau$-bimonad in $\K$.

In view of the above said, all the trivalent Hopf data from Table \ref{table:2} are paired wreaths. Consequently, in the latter cases a Hopf datum is equivalent to a paired wreath.
\end{thm}

The Theorem applies also to the monad Hopf datum studied in \prref{simplest case}, which yields the following four Hopf data which simultaneously are paired wreaths:
\begin{table}[h!]
\begin{center}
\begin{tabular}{ c  c }
 $(\psi, \sigma)$ &  (matching cowreaths) \\ [0,5ex]
\hline
 $ ((0,0),0)$ & \hspace{0,4cm} $((\crta 0,\crta 0),\crta 0), ((\crta 1,\crta 0),\crta 0), ((\crta 0,\crta 1),\crta 0), ((\crta 1,\crta 1),\crta 0)$  \\ [1ex]
\end{tabular}
\end{center}
\end{table}

Observe that the three cases in the third line in Table \ref{table:2} are $\alpha$-symmetric to the corresponding ones in the second line. So in total there are 2+4+3+4=13 non-isomorphic
trivalent paired wreaths.

\subsection{Examples}

When $\K=\hat\C$ where $\C$ is a braided monoidal category, the r\^ole of the 2-cells $\tau$ is played by the braiding. If $\C$ is not braided, $\tau$ is a local braiding,
existing between the objects $B$ and $F$ and all their combinations (recall the (co)monadic distributive law properties, Yang-Baxter equations and naturalities for $\tau$).

\begin{ex}
When $\K=\hat\C$, where $\C$ is a monoidal category, a paired wreath and a Hopf datum are written out in the string diagrams on the previous pages, where the strings (2-cells) are morphisms,
their source and targets (1-cells) are objects, and $B$ is an algebra, $F$ a coalgebra in $\C$, satisfying the listed axioms.
\end{ex}

\begin{ex}
The paired wreath $((1,0),0), ((\crta 1,\crta 0),\crta 0)$ is a 2-categorical version of Radford biproduct bialgebra.
Set $\K=\hat Vec$.
Then \cite[Theorem 2.1 and Proposition 2]{Rad1}, that characterizes when Radford biproduct $B\times F$ is a bialgebra, is a special case of \thref{sufficient for bimonad} in
the setting $((1,0),0), ((\crta 1,\crta 0),\crta 0)$, {\em i.e.}  $(\psi_1, \phi'_2)$. This 2-categorical formulation of the Radford biproduct coincides with our
biwreath from \cite{Femic5}.
\end{ex}

\begin{ex} \exlabel{ex 2}
The paired wreath $((1,0),0), ((\crta 0,\crta 1),\crta 0)$ is a 2-categorical version of Majid's bicrossproduct.
Similarly, as in the above example, in $\K=\hat Vec$ there is \cite[Theorem 3.3]{Maj5}, which characterizes when Majid's bicrossproduct is a bialgebra.
It is a special case of \thref{sufficient for bimonad} in
the setting $((1,0),0), ((\crta 0,\crta 1),\crta 0)$, {\em i.e.}  $(\psi_1, \phi'_1)$ (or $\alpha$-symmetrically: $((0,1),0), ((\crta 1,\crta 0),\crta 0)$, {\em i.e.}  $(\psi_2, \phi'_2)$).
\end{ex}

Observe that since in the above two examples $\sigma$ and $\rho'$ are trivial, {\em i.e.} the 2-cels $\mu_M$ and $\Delta_C'$ are canonical, from the bialgebra condition on
$B\times F$ one may deduce the compatibility conditions for $F$ in \equref{(co)unital} and \equref{(co)unital mixed}, as well as \equref{psi unital add} (or the compatibility
conditions for the (co)actions in \coref{(co)actions}). In a general setting, in a paired wreath though, we are forced to require these compatibility conditions as part of the definition.

\begin{ex}
The wreath $((0,1),1)$ in general $\K$ is a 2-categorical generalization of Sweedler's cocycle twisted smash product algebra, \cite{Sw1}. Moreover, it is the 2-categorical formulation
of the biwreath-like object we studied in \cite{Femic5} for $\K=\hat\C$, where $\C$ is a braided monoidal category.
Its $\pi$-symmetric cowreath is the dual construction representing Sweedler's cycle twisted cosmash coproduct coalgebra.
\end{ex}

\begin{ex}
At the end of \cite{Maj6} Majid included {\em cocycle bicrossproducts} which are bialgebras in $Vec$. They correspond to $((0,1),1), ((\crta 1,\crta 0),\crta 1)$
in our setting. Namely, Majid's conditions C, A, B, D correspond to our first and second identities in \equref{1-3} and \equref{4-6}, respectively.
(To see the latter one of the four, write out our identity and apply \equref{F mod alg} and \equref{B comod coalg}.)
In Table \ref{table:2} we see that in the 2-categorical setting it only remains to prove that $FB$ is a bimonad in order to be able to claim that $(B,F)$ is
a paired wreath.
\end{ex}

\begin{ex}
In \cite[Proposition 3.12]{Maj5} Majid characterized a {\em matched pair of bialgebras} $(B,F)$ in $Vec$. It turns out to be a Hopf datum for $\K=\hat Vec$ in the
setting $((1,1),0), ((\crta 0,\crta 0),\crta 0)$, {\em i.e.}  $(\psi_4, \tau_{F,B})$. The ``fourth bialgebra compatibility'' for $B$ and $F$ are our identities
\equref{proj B-bialg} and \equref{proj F-bialg}, left and right module coalgebra assumptions are our second and third identity in \equref{1-3}, and Majid's conditions
A, B, C are our identities \equref{mod alg}, \equref{F mod alg} and the first identity in \equref{1-3}, respectively. The rest of our identities in a Hopf datum hold trivially,
since the two coactions and the (co)cycles are trivial.

Matched pairs of bialgebras in any braided monoidal category $\C$ are treated in \cite{BD1, ZC, Femic1}. In \cite[Theorem 1.4]{ZC} it is proved that given a
matched pair of bialgebras $(B,F)$ in $\C$ the ``wreath'' product $B\times F$ (usually called {\em double cross product} in this setting) is a bialgebra in $\C$.
Recall that the Drinfel`d double is a particular case.

\thref{sufficient for bimonad} confirms the result from the 2-categorical viewpoint.
\end{ex}

\begin{ex} \exlabel{ex 6}
One of the Hopf data for which we could not prove to be a paired wreath (neither a wreath) is determined by $((1,1),1), ((\crta 1,\crta 0),\crta 0)$, {\em i.e.} $(\psi_4, \phi'_2)$.
If $\K=\hat Vec$, this Hopf datum is Schauenburg's cosmash product defined in \cite[Theorem 5.1]{Sch3}.
\end{ex}

We end this subsection proving the following result, which is a generalization of \cite[Lemma 5.2]{Femic5}.

\begin{lma} \lelabel{lambda d.l.}
In a paired wreath and a Hopf datum determined by $((1,0),0), ((\crta 1,\crta 0),\crta 0)$, {\em i.e.} Radford biproduct in $\K$,
the 1-cell $B$ is a right bimonad, and $F$ is a left bimonad in $\K$.
\end{lma}

\begin{proof}
In view of the identity \equref{proj F-bialg} it suffices to prove that the 2-cell $\lambda_F$ from \equref{lambda F}
is left monadic and comonadic distributive law. The claim for $B$ will then hold by $\pi$-symmetry. We find:
$$\hspace{-0,2cm}
\gbeg{3}{5}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\gmu \gcn{1}{1}{1}{0} \gnl
\gvac{1} \hspace{-0,34cm} \glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gnl
\gvac{1} \gcl{1} \gcl{1} \gnl
\gvac{1} \gob{1}{F} \gob{1}{F}
\gend=
\scalebox{0.84}[0.84]{
\gbeg{5}{9}
\got{1}{F} \gvac{1} \got{1}{F}\got{1}{F}\gnl
\gwmu{3} \gcl{4} \gnl
\gvac{1} \gcl{1} \gnl
\gwcm{3} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gnl
\gcl{1} \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{1} \glm \gcl{2} \gnl
\gwmu{3} \gnl
\gvac{1} \gob{1}{F} \gvac{1} \gob{1}{F}
\gend}
\stackrel{\equref{proj F-bialg}}{=}
\scalebox{0.84}[0.84]{
\gbeg{7}{10}
\gvac{1} \got{1}{F} \gvac{1} \got{2}{F} \got{1}{F}\gnl
\gwcm{3} \gcmu \gcl{4} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \gcl{2} \gnl
\gcl{1} \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{1} \glm \gmu \gnl
\gwmu{3} \hspace{-0,22cm} \grmo \gvac{1} \gcl{1} \gcn{1}{1}{2}{1} \gnl
\gvac{2} \gcn{1}{1}{0}{1} \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gvac{2} \gcl{1} \glm \gcl{2} \gnl
\gvac{2} \gwmu{3} \gnl
\gvac{3} \gob{1}{F} \gob{3}{F}
\gend}\stackrel{ass. F}{\stackrel{*}{=}}
\scalebox{0.84}[0.84]{
\gbeg{8}{13}
\gvac{1} \got{1}{F} \gvac{2} \got{1}{F} \got{3}{F}\gnl
\gwcm{3} \gwcm{3} \gcl{7} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \gvac{1} \gcl{2} \gnl
\gcl{1} \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{6} \glm \gcn{1}{1}{1}{2} \gcn{1}{1}{3}{4} \gnl
\gvac{2} \gcl{5} \hspace{-0,22cm} \grmo \gvac{1} \gcl{1} \grmo \gvac{1} \gcl{1} \gnl
\gvac{3} \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gvac{3} \gmu \gmu \gnl
\gvac{4} \gcn{1}{1}{0}{2} \gvac{1} \hspace{-0,34cm} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gvac{5} \glm \gcl{3} \gnl
\gcn{1}{1}{3}{4} \gvac{2} \gwmu{4} \gnl
\gvac{2} \hspace{-0,34cm} \gwmu{4} \gnl
\gvac{3} \gob{2}{F} \gob{6}{F}
\gend}$$

$$\stackrel{B\x comod.}{\stackrel{B\x mod.}{=}}
\scalebox{0.84}[0.84]{
\gbeg{9}{13}
\gvac{1} \got{2}{F} \gvac{2} \got{1}{F} \got{3}{F}\gnl
\gwcm{4} \gwcm{3} \gcl{4} \gnl
\gcl{1} \gvac{1} \grmo \gvac{1} \gcl{1} \gcl{1} \grmo \gvac{1} \gcl{1} \gnl
\gcl{1} \gcn{1}{1}{3}{2} \gvac{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gcl{2} \gnl
\gcl{7} \gcmu \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gvac{1} \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gmu \gcn{1}{1}{1}{0} \gnl
\gvac{1} \glm \gcl{1} \gcn{1}{1}{1}{2} \gvac{1} \hspace{-0,34cm} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gvac{2} \gcn{1}{2}{2}{2} \gcn{1}{2}{2}{5} \gvac{1} \glm \gcl{5} \gnl
\gvac{6} \gcl{1} \gnl
\gvac{2} \gcn{1}{1}{2}{3} \gvac{2} \glm \gcl{3} \gnl
\gvac{3} \gwmu{4} \gnl
\gvac{1} \hspace{-0,34cm} \gwmu{5} \gnl
\gvac{3} \gob{2}{F} \gob{6}{F}
\gend}
\stackrel{\tau}{\stackrel{nat.}{\stackrel{\equref{mod alg}}{=}}}
\scalebox{0.84}[0.84]{
\gbeg{8}{10}
\gvac{1} \got{1}{F} \gvac{2} \got{1}{F} \got{3}{F}\gnl
\gwcm{3} \gwcm{3} \gcl{2} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \grmo \gvac{1} \gcl{1} \gnl
\gcl{1} \gcl{3} \gbr \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{4} \gcl{1} \gcl{2} \gcn{1}{1}{1}{3} \glm \gcl{1} \gnl
\gvac{4} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gcl{1} \gnl
\gvac{1} \gcn{1}{1}{1}{3} \gwmu{3} \gmu \gnl
\gvac{2} \glm \gcn{1}{2}{4}{4} \gnl
\gwmu{4} \gnl
\gvac{1} \gob{2}{F} \gob{6}{F}
\gend}
\stackrel{\tau}{=}
\scalebox{0.84}[0.84]{
\gbeg{7}{12}
\gvac{1} \got{1}{F} \gvac{1} \got{1}{F} \got{3}{F}\gnl
\gvac{1} \gcl{5} \gwcm{3} \gcl{2} \gnl
\gvac{2} \gcl{1} \grmo \gvac{1} \gcl{1} \gnl
\gvac{2} \gcl{1} \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gvac{2} \gcl{1} \glm \gcl{6} \gnl
\gvac{2} \gwmu{3} \gnl
\gwcm{3} \gcl{2} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gnl
\gcl{1} \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{1} \glm \gcl{1} \gnl
\gwmu{3} \gwmu{3} \gnl
\gvac{1} \gob{1}{F} \gob{5}{F}
\gend}=
\gbeg{4}{5}
\got{1}{F} \got{1}{F} \got{1}{F} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gnl
\glmptb \gnot{\hspace{-0,34cm}\lambda_F} \grmptb \gcl{1} \gnl
\gcl{1} \gmu \gnl
\gob{1}{F} \gob{2}{F.}
\gend
$$
At the place * we applied the first identity in \equref{4-6}. Observe that the pre-multiplication of $F$ is associative by \equref{weak assoc. mu_M} since $\sigma$ is trivial.
The left $B$-comodule structure on $F$ is proper, since $\rho'$ is trivial, see \equref{weak coaction}. In the fifth equality we applied naturality of $\tau_{F,F}$ with respect to
$\gbeg{2}{1}
\glm \gnl
\gend$ and the monadic distributive law of $\tau_{F,F}$, a part from \equref{mod alg}, and the same distributive law we applied in the sixth equality. The (co)unital
distributive law property for $\lambda$ we proved before \equref{mu-delta-final}. The resting comonadic distributive law property follows by vertical symmetry:
one uses \equref{proj F-bialg}, the coassociativity of the comultiplication of $F$, the third identity in \equref{1-3}, left $B$-(co)module structures of $F$, comonadic
distributive law property of $\tau_{F,F}$, naturality of $\tau_{F,F}$ with respect to
$\gbeg{2}{1}
\grmo \gvac{1} \gcl{1} \gnl
\gend$
and \equref{B comod coalg}.
\qed\end{proof}

\subsection{New definitions arising from Hopf data and concluding remarks}

The names we put in the identities \equref{mod alg}--\equref{normalized 2-cycle ro'} suggest new definitions of the respective objects, more precisely, 1-cells in 2-categories.
We highlight some of them here. Given a monad Hopf datum $(B,F, \psi, \mu_M)$ in $\K$ (we consider $\eta_M$ canonical), the following notions are defined through the corresponding identity:
\begin{enumerate}
\item a (normalized) 2-cocycle,
\item action twisted by a 2-cocycle,
\item module monad (in a broader sense: the monad is weak associative, or the action on the monad is twisted by a 2-cocycle).
\end{enumerate}
A comonad Hopf datum delivers $\pi$-symmetric notions.

\medskip

Observe that taking into account the formula \equref{mu-delta-final} for $\mu_M$, weak associativity of the pre-multiplication on $F$ (\equref{weak assoc. mu_M}),
the 2-cocycle condition and twisted action get their specific forms:
$$
\gbeg{4}{4}
\got{1}{F} \got{1}{F} \got{3}{F} \gnl
\gcl{1} \gwmu{3} \gnl
\gwmu{3} \gnl
\gob{3}{F}
\gend=
\gbeg{5}{8}
\got{1}{} \got{1}{F} \gvac{1} \got{2}{F} \got{1}{F} \gnl
\gwcm{3} \gcmu \gcl{5} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \gcl{2} \gnl
\gcl{1} \gcl{1} \gbr \gnl
\gcl{1} \glm \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb  \gnl
\gwmu{3} \gvac{1} \glm \gnl
\gvac{1} \gwmu{5} \gnl
\gob{7}{F}
\gend
$$
\begin{center}
{\footnotesize weak associativity}
\end{center}

$
\gbeg{6}{9}
\got{1}{F} \got{3}{F} \got{2}{F}\gnl
\gcl{5} \gwcm{3} \gcmu \gnl
\gvac{1} \gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \gcl{2} \gnl
\gvac{1} \gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gvac{1} \gcl{1} \glm \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb  \gnl
\gvac{1} \gwmu{3} \gvac{1} \gcl{2} \gnl
\glmpt \gnot{\sigma} \gcmp \grmptb  \gnl
\gvac{2} \gwmu{4} \gnl
\gob{8}{B}
\gend=
\gbeg{6}{11}
\got{3}{F} \got{2}{F} \got{2}{F} \gnl
\gwcm{3} \gcmu \gcn{1}{4}{2}{2} \gnl
\gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \gcl{2} \gnl
\gcl{1} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{F,F}} \grmptb \gnl
\gcl{1} \glm \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmpt  \gnl
\gwmu{3} \gcmu \gcmu \gnl
\gvac{1} \gcn{1}{2}{1}{3} \gvac{1} \gcl{1} \glmpt \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\gvac{3} \glm \grmo \gcl{1} \gnl
\gvac{2} \glmpt \gnot{\sigma} \gcmpb \grmpt \gcl{1} \gnl
\gvac{3} \gwmu{3} \gnl
\gob{9}{B}
\gend
\hspace{1,5cm}
\gbeg{6}{11}
\got{2}{B} \got{2}{F} \got{2}{F} \gnl
\gcmu \gcmu \gcn{1}{4}{2}{2} \gnl
\gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1} \gnl
\glm \grmo \gcl{1} \gnl
\gvac{1} \gcl{4} \gcn{1}{1}{1}{2} \gnl
\gvac{2} \gcmu \gcmu \gnl
\gvac{2} \gcl{1} \glmptb \gnot{\hspace{-0,34cm}\tau_{B,F}} \grmptb \gcl{1}  \gnl
\gvac{2} \glm \grmo \gcl{1}  \gnl
\gvac{1} \glmpt \gnot{\sigma} \gcmpb \grmpt \gcl{1} \gnl
\gvac{2} \gwmu{3} \gnl
\gob{7}{B}
\gend=
\gbeg{3}{10}
\got{1}{B} \gvac{1} \got{1}{F} \gvac{1} \got{2}{F} \gnl
\gcl{6} \gwcm{3} \gcmu \gnl
\gvac{1} \gcl{1} \grmo \gvac{1} \gcl{1} \gcl{1} \gcl{2} \gnl
\gvac{1} \gcl{1} \gcl{1} \gbr \gnl
\gvac{1} \gcl{1} \glm \glmpt \gnot{\hspace{-0,34cm}\sigma} \grmptb  \gnl
\gvac{1} \gwmu{3} \gvac{1} \gcl{3}  \gnl
\gvac{1} \gcn{1}{1}{3}{1}  \gnl
\grmo \gcl{1} \gnl
\gwmu{6} \gnl
\gob{6}{B}
\gend
$$ \vspace{0,4cm}
\hspace{3,6cm} {\footnotesize 2-cocycle condition} \hspace{4,2cm} {\footnotesize twisted action}

\noindent This corresponds to the monad Hopf datum $((1,1),1)$.
When $\K=\hat Vec$ this 2-cocycle condition and twisted action read:
\begin{multline}
 \sigma(f,g_{(1)} (g_{(2)_{[-1]}}\rtr k_{(1)})) \sigma(g_{(2)_{[0]}}, k_{(2)}) = \\
\sigma(f_{(1)}(f_{(2)_{[-1]}}\rtr g_{(1)}), (\sigma(f_{(2)_{[0]}}, g_{(2)})_{(1)}\rtr k_{(1)})) (\sigma(f_{(2)_{[0]}}, g_{(2)})_{(2)}\ltr k_{(2)})
\end{multline}
and
\begin{multline}
 \sigma(b_{(1)}\rtr f_{(1)}, (b_{(2)}\ltr f_{(2)})_{(1)}\rtr g_{(1)}) ((b_{(2)}\ltr f_{(2)})_{(2)}\ltr g_{(2)}) = \\
(b\ltr (f_{(1)}(f_{(2)_{[-1]}}\rtr g_{(1)}))) \sigma(f_{(2)_{[0]}}, g_{(2)})
\end{multline}
for $f,g,k\in F, b\in B$, respectively.
When the left $B$-action on $F$ is trivial, this (more general) definition of a 2-cocycle and twisted action recovers the twisted action by Sweedler's 2-cocycle.

\bigskip

The identities \equref{1-3} and \equref{4-6} holding in a Hopf datum (and a paired wreath) suggest some new definitions, too. Recall, as we observed in \rmref{when cocycles are trivial},
that when $\sigma$ and $\rho'$ are trivial, the second and the fourth of these six identities are $\alpha$-symmetric to the third and the sixth one, respectively,
and the fifth identity is trivially fulfilled.
Then we obtain new definitions of a Yetter-Drinfel`d condition and module comonad and comodule monad in the obvious way. Observe that these structures appeared in \exref{ex 2}.
But we also may consider module comonads or comodule monads with non-trivial (co)cycles, as they appear in \exref{ex 6}.

\bigskip

The benefit of having defined paired wreaths and Hopf data in 2-categories is that one may study different types of crossed (co)products in much larger class of cases, taking for
$\K$ to be any 2-category that one may pick up. From those
whose 0-cells are certain elements from some vector space, to those whose 0-cells are proper 2-categories, just to mention some of them.

\medskip

As we pointed out, one may use generalized definitions of 2-cocycles, twisted actions and (co)module (co)algebras
and study in this new setting the known constructions which are done with the classical definitions. For example, the Brauer group of $H$-module algebras, for a quasi-triangular Hopf algebra
$H$ over a field, Galois objects, etc.

redpajama_set_name: RedPajamaArXiv