text:
\section{Introduction}

A  $\cC$-valued equivariant coarse homology theory is a functor $$E:G\mathbf{BornCoarse}\to \cC$$ satisfying a certain family of axioms  {\cite[Def. 3.10]{equicoarse}}.  Here, $G\mathbf{BornCoarse}$ is the category of $G$-bornological coarse spaces and $\cC$ is a stable cocomplete $\infty$-category.
We refer to \cite[Sec. 2.1]{equicoarse},  or {Section~\ref{geriogjheroigfwefwefw},} for details. The category $G\mathbf{BornCoarse}$ has a symmetric monoidal structure
$\otimes$, and if {also} $\cC$ has a symmetric monoidal structure, then we can ask whether {the functor} $E$ can be refined to a
lax symmetric monoidal functor.   Such a refinement can simplify calculations or can be applied to {obtain} localization results, see  \cite{bulu}.

In the present paper{, as an example for  $E$,} we consider the universal coarse algebraic $K$-homology  $$\mathrm{UK}\cX_{{\mathbf{A}}}^{G}:G\mathbf{BornCoarse}\to \cM_{loc}$$ associated to an additive category ${\mathbf{A}}$ with a strict action of the group $G$, where $\cM_{loc}$ is the {stable $\infty$-category on non-commutative motives, defined as the target of the} universal localizing invariant {${\mathcal{U}}_{loc}$} of Blumberg-Gepner-Tabuada \cite{MR3070515}. The functor $\mathrm{UK}\cX_{{\mathbf{A}}}^{G}$
has been introduced in \cite{buci} as the universal variant of the spectrum-valued
coarse algebraic $K$-homology  $K\cX_{{\mathbf{A}}}^{G}$ constructed in \cite[Ch. 8]{equicoarse}.

By \cite[Thm. 5.8]{bgt-2}, the {$\infty$-}category $\cM_{loc}$ has a symmetric monoidal structure.

The main result of the present paper is the following theorem.
\begin{theorem}\label{rgreioghjo34tergergeg} A symmetric monoidal structure on  ${\mathbf{A}}$  induces a  {lax} symmetric monoidal refinement
of the  functor $\mathrm{UK}\cX_{{\mathbf{A}}}^{G}$.
\end{theorem}

The functor $\mathrm{UK}\cX_{{\mathbf{A}}}^{G}$ is constructed as a composition of
functors
$$\mathrm{UK}\cX_{{\mathbf{A}}}^{G}:G\mathbf{BornCoarse}\stackrel{\bV_{{\mathbf{A}}}^{G}}{\to} {\mathtt{Add}}_{1}\stackrel{\mathrm{UK}}{\to} \cM_{loc}\ ,$$
where $X\mapsto \bV_{{\mathbf{A}}}^{G}(X)$ associates to a $G$-bornological coarse space $X$ its additive category of  equivariant $X$-controlled ${\mathbf{A}}$-objects {(see Definition \ref{def:Xcontrolledobject})}, and $\bC\mapsto \mathrm{UK}(\bC)$ sends an additive category $\bC$ to
the motive ${\mathcal{U}}_{loc}(\Ch^{b}(\bC)[W^{-1}])$ of the associated stable $\infty$-category
$\Ch^{b}(\bC)[W^{-1}]$ of bounded chain complexes over $\bC$, where $W$ is the set  of homotopy equivalences.
Since $\mathrm{UK}$ sends equivalences of additive categories to equivalences it has a factorization
$$\mathrm{UK}:{\mathtt{Add}}_{1}\stackrel{loc}{\to} {\mathtt{Add}}_{\infty}\stackrel{\mathrm{UK}_{\infty}}{\to} \cM_{loc} \ ,$$
where $loc:{\mathtt{Add}}_{1}\to {\mathtt{Add}}_{\infty}:={\mathtt{Add}}_{1}[W^{-1}_{{\mathtt{Add}}}]$ is the localization at the equivalences {$W_{{\mathtt{Add}}}$ of additive categories,} in the realm of $\infty$-categories.

Theorem \ref{rgreioghjo34tergergeg} follows from the following two assertions:

\begin{theorem}[{Theorem \ref{mainthm}}]\label{rgreioghjo34tergergeg1}
A symmetric monoidal structure on ${\mathbf{A}}$  induces a {lax} symmetric monoidal refinement
$$\bV_{{\mathbf{A}}}^{G,\otimes}:G\mathbf{BornCoarse}\to {\mathtt{Add}}_{\infty}^{\otimes}$$
of the functor
$loc\circ \bV_{{\mathbf{A}}}^{G}$.
\end{theorem}

\begin{theorem}[Theorem \ref{rgreioghjo34tergergeg21}] \label{rgreioghjo34tergergeg2}
The functor $\mathrm{UK}_{\infty}$ admits a  {lax} symmetric monoidal refinement
$$\mathrm{UK}_{\infty}^{\otimes}:{\mathtt{Add}}_{\infty}^{\otimes}\to \cM_{loc}^{\otimes}\ .$$
\end{theorem}

The main difficulty in {proving} Theorem \ref{rgreioghjo34tergergeg1} is that the symmetric monoidal category of {small} additive categories is of a $2$-categorical nature. A {pedestrian} approach to {the proof of} this theorem would thus require to work with symmetric monoidal structures on $2$-categories and therefore   tedious considerations of a large set of commuting diagrams.  In this paper we prefer to use the language of symmetric monoidal $\infty$-categories.
In Section \ref{vevgebvveerverv}{, by using  the {Grothendieck} construction,}
 we encode the functor $\bV_{{\mathbf{A}}}^{G}{\colon G\mathbf{BornCoarse} \to {\mathtt{Add}}_{1}}$ into {a} cocartesian fibration $\cV_{{\mathbf{A}}}^{G}\to G\mathbf{BornCoarse}$
coming from an op-fibration of $1$-categories. We then encode a symmetric monoidal
refinement of the functor $\bV_{{\mathbf{A}}}^{G}$ into a symmetric monoidal  structure on $\cV_{{\mathbf{A}}}^{G}$ and a
symmetric monoidal refinement of  the functor to $G\mathbf{BornCoarse}$. This only requires $1$-categorical considerations.
The machine of $\infty$-categories then produces, as explained in Section \ref{fgoiwfwefewfwefwefwe},  the asserted  symmetric monoidal refinement
in Theorem \ref{rgreioghjo34tergergeg1}.

The technical  results  Theorem  \ref{gui34tefrgeg} and Theorem  \ref{rgio34t34r34rerg} might be of independent interest in cases where one wants to construct symmetric monoidal refinements of functors from $1$-categories to ${\mathbf{Cat}}_{1}$ or ${\mathtt{Add}}_{1}$.

Theorem \ref{rgreioghjo34tergergeg2} is shown in Section \ref{foiwejfowfwefefwef} by combining   various results  in the literature on $dg$-categories.

 {\em Acknowledgements: We thank Denis-Charles Cisinksi and Thomas Nikolaus for helpful discussion. U.B. was supported by the SFB 1085 (Higher Invariants) and L.C. was {supported} by the GK 1692 (Curvature, Cycles, and Cohomology).}

\section{Symmetric monoidal functors to ${\mathbf{Cat}}$ and ${\mathtt{Add}}$}\label{fgoiwfwefewfwefwefwe}

{In this section, we construct lax symmetric monoidal refinements of functors from symmetric monoidal $1$-categories to the categories ${\mathbf{Cat}}_{1}$  (and ${\mathtt{Add}}_{1}$) of small (additive) categories. }

\subsection{From $2$- to $\infty$-categories}

 A symmetric monoidal structure on a $1$-category $\bC$ consists of the tensor functor $$\otimes_{\bC}:\bC\times \bC\to \bC\ ,$$ the tensor unit $1_{\bC}$, and the associator, symmetry and unit-transformations, which must satisfy various compatibility relations.
 If $\bC$ and $\bD$  are   symmetric monoidal  $1$-categories, then we can consider lax symmetric monoidal functors from $\bC$ to $\bD$.
Such a lax symmetric monoidal  functor is given by  a functor  $F:\bC\to \bD$     together with
a natural transformation $$F(C)\otimes_{\bD} F(C')\to F(C\otimes_{\bC} C^{\prime})\ , \quad C,C^{\prime}\in \bC$$
{that} is compatible with the associators, symmetries and unit-transformations of $\bC$ and $\bD$ in a suitable way. We will list these structures and relations in Subsection \ref{ewfoiewfwefwefwef} below.

The categories ${\mathtt{Add}}$ or ${\mathbf{Cat}}$ of small additive categories and small categories are naturally $2$-categories.  Furthermore, the category ${\mathbf{Cat}}$ is symmetric monoidal with respect to the Cartesian symmetric monoidal structure $\times:=\otimes_{{\mathbf{Cat}}}$. The category ${\mathtt{Add}}$ has also a symmetric monoidal structure  $\otimes_{{\mathtt{Add}}}$: if ${\mathbf{A}}$ and ${\mathbf{B}}$ are two additive categories, then the objects of the tensor product ${\mathbf{A}}\otimes_{{\mathtt{Add}}} {\mathbf{B}}$
are pairs $(A,B)$ of objects   $A$ in ${\mathbf{A}}$ and $B$ in ${\mathbf{B}}$, and the morphisms are given by the tensor product
$$\Hom_{{\mathbf{A}}\otimes_{{\mathtt{Add}}} {\mathbf{B}}}((A,B),(A^{\prime},B^{\prime})):= \Hom_{{\mathbf{A}}}(A,A^{\prime})\otimes_{\Z} \Hom_{{\mathbf{B}}}(B,B^{\prime})$$ of abelian groups.

In the case of a  symmetric monoidal structure on a $2$-category, like ${\mathbf{Cat}}$ or ${\mathtt{Add}}$, {we have the same {compatibility} relations} between the structures {(tensor functor, tensor unit,  etc.)} {as} in the $1$-categorical case, {but they} are satisfied up to $2$-morphisms {only,} which in turn must satisfy higher compatibility relations. A similar remark applies to the notion of a (lax) symmetric monoidal functor.

In the present paper we consider the $1$-categorical situation as explicitly manageable, and we will avoid to explicitly work with symmetric monoidal structures on $2$-categories.

Let $\bC$ be a symmetric monoidal $1$-category. Our goal is to construct symmetric monoidal functors
$F:\bC\to {\mathbf{Cat}}$ or $F:\bC\to {\mathtt{Add}}$ using $1$-categorical data only. Instead of working with the symmetric monoidal $2$-categories ${\mathbf{Cat}}$ or ${\mathtt{Add}}$ we will actually use the associated symmetric monoidal
$\infty$-categories ${\mathbf{Cat}}_{\infty}$ or  ${\mathtt{Add}}_{\infty}$.

We  start with the
 ordinary category ${\mathbf{Cat}}_{1}$ of small categories.  Let $W_{{\mathbf{Cat}}}$ be the equivalences in $ {\mathbf{Cat}}_{1} $. The localization in large $\infty$-categories
$${\mathbf{Cat}}_{\infty}:= \Nerve({\mathbf{Cat}}_{1}) [W_{{\mathbf{Cat}}}^{-1}]$$
is the large $\infty$-category of categories. It models the $2$-category ${\mathbf{Cat}}$ in the following sense.
The $2$-category ${\mathbf{Cat}}$  can be considered as a category enriched in categories.
Applying the nerve functor {$\Nerve$} to the $\Hom$-categories in ${\mathbf{Cat}}$
we get a fibrant\footnote{i.e.,  the $\Hom$-complexes are Kan complexe
} simplicially enriched category $\Nerve({\mathbf{Cat}})$. Applying the {homotopy} coherent nerve functor {$\mathcal{N}$,} we get an $\infty$-category
$$\Nerve_{2}({\mathbf{Cat}}):=\mathcal{N}(\Nerve({\mathbf{Cat}}))\ . $$
Then, we have an equivalence of $\infty$-categories
$$\Nerve_{2}({\mathbf{Cat}})\simeq {\mathbf{Cat}}_{\infty}\ . $$
We refer to the appendix of \cite{GHN17} for more details about $\Nerve_{2}$.

The category ${\mathbf{Cat}}_{1}$ is a symmetric monoidal category and therefore gives rise to an op-fibration   of $1$-categories \cite[Constr. 2.0.01]{HA}, and {to} a symmetric monoidal $\infty$-category  \cite[Def. 2.0.0.7 \& Ex. 2.1.2.21]{HA}
$${\mathbf{Cat}}_{1}^{\otimes}\to \mathbf{Fin}_{*}\ ,\text{ and }\quad \Nerve({\mathbf{Cat}}_{1}^{\otimes})\to \Nerve(\mathbf{Fin}_{*}) \ ,$$  respectively.
The equivalences $W_{{\mathbf{Cat}}}$ are preserved by the cartesian product. Hence we can form a symmetric monoidal localization \cite[Prop. 3.2.2]{hinich}     $${\mathbf{Cat}}_{\infty}^{\otimes} :=\Nerve({\mathbf{Cat}}_{1}^{\otimes})[W_{{\mathbf{Cat}}}^{\otimes,-1}] \to \Nerve(\mathbf{Fin}_{*})$$
 whose underlying $\infty$-category is equivalent to ${\mathbf{Cat}}_{\infty}$. Conseqently,   the symmetric monoidal $\infty$-category ${\mathbf{Cat}}_{\infty}^{\otimes}\to  \Nerve(\mathbf{Fin}_{*})$
 models the symmetric monoidal $2$-category ${\mathbf{Cat}}$. In this way we avoid to spell out the structures of a symmetric monoidal $2$-category explicitly.

A similar reasoning applies to ${\mathtt{Add}}$. We consider the {large} $1$-category ${\mathtt{Add}}_{1}$ of {small} additive categories and exact functors with the equivalences $W_{{\mathtt{Add}}}$. Then we define  the {large} $\infty$-category
$${\mathtt{Add}}_{\infty}:=\Nerve({\mathtt{Add}}_{1})[W_{{\mathtt{Add}}}^{-1}]$$ and get an equivalence
$${\mathtt{Add}}_{\infty}\simeq \Nerve_{2}({\mathtt{Add}})\ .$$
We can consider ${\mathtt{Add}}_{1}$ as a symmetric monoidal category giving rise to {an}  op-fibration of $1$-categories and  a symmetric monoidal $\infty$-category
$${\mathtt{Add}}_{1}^{\otimes} \to \mathbf{Fin}_{*}\ , \quad \Nerve({\mathtt{Add}}_{1}^{\otimes}) \to \Nerve(\mathbf{Fin}_{*})\ .$$
Since the equivalences $W_{{\mathtt{Add}}}$  are   preserved by the tensor  product {$\otimes_{{\mathtt{Add}}}$,} we get the
symmetric monoidal localization
 $${\mathtt{Add}}_{\infty}^{\otimes}:=\Nerve({\mathtt{Add}}_{1}^{\otimes})[W_{{\mathtt{Add}}}^{\otimes,-1}] \to \Nerve(\mathbf{Fin}_{*})$$
 whose underlying $\infty$-category is equivalent to ${\mathtt{Add}}_{\infty}$. Therefore  ${\mathtt{Add}}_{\infty}^{\otimes}\to \Nerve(\mathbf{Fin}_{*})$
  models the symmetric monoidal $2$-category ${\mathtt{Add}}$.

Let $\bC$ be an ordinary category.
A functor $F:\bC\to {\mathbf{Cat}}_{1}$ (or $F:\bC\to {\mathtt{Add}}_{1}$) gives rise to a functor between $\infty$-categories
$F_{\infty}:\Nerve(\bC)\to  {\mathbf{Cat}}_{\infty}$ (or $F_{\infty}:\Nerve(\bC)\to {\mathtt{Add}}_{\infty}$) in the natural way, e.g. as the composition
$$F_{\infty}: \Nerve(\bC)\stackrel{\Nerve(F)}{\to} \Nerve({\mathbf{Cat}}_{1})\to    \Nerve({\mathbf{Cat}}_{1})[W_{{\mathbf{Cat}}}^{-1}]={\mathbf{Cat}}_{\infty}\ .$$

A symmetric monoidal $1$-category $\bC$ gives rise to  the symmetric monoidal $\infty$-category  $\Nerve(\bC^{\otimes})\to \Nerve(\mathbf{Fin}_{*})$ whose underlying $\infty$-category is equivalent to $\Nerve(\bC)$. We now consider a functor
 $F:\bC\to {\mathbf{Cat}}_{1}$ (or $F: \bC\to {\mathtt{Add}}_{1}$). {Recall that a map of $\infty$-operads \cite[Def. 2.1.2.7]{HA} can be thought of as a (lax) symmetric monoidal functor \cite[Def. 2.1.3.7]{HA} between the underlying categories.}

Let $F$ and $F_{\infty}$ be as above.

 \begin{ddd}\label{gfuzfkuzfkuzf}
 	A lax symmetric monoidal refinement of  $ F$
	is   a morphism of $\infty$-operads
$$F^{\otimes}:\Nerve(\bC^{\otimes})\to {\mathbf{Cat}}^{\otimes}_{\infty}\ , \quad (F^{\otimes}:\Nerve(\bC^{\otimes})\to {\mathtt{Add}}^{\otimes}_{\infty})$$
{that} induces a functor equivalent to $F_{\infty}$ on the  underlying $\infty$-categories.
\end{ddd}

Using this definition we avoid to spell out the details of the notion of a lax-symmetric functor from $\bC$ to the $2$-category ${\mathbf{Cat}}$ or ${\mathtt{Add}}$.

\subsection{Symmetric monoidal refinements of functors to ${\mathbf{Cat}}_{1}$ and ${\mathtt{Add}}_{1}$}

{In this subsection we state the technical results Theorem \ref{gui34tefrgeg} and Theorem \ref{rgio34t34r34rerg} which provide lax symmetric monoidal refinements of functors to ${\mathbf{Cat}}_{1}$ and ${\mathtt{Add}}_{1}$.}

Let $\bC$ be a $1$-category. A functor between  $1$-categories
$$F:\bC\to {\mathbf{Cat}}_{1}$$ can be interpreted, via the Grothendieck construction,
as a cocartesian fibration
$$\pi_{F}:\cF\to \bC\ .$$ An object of the $1$-category
$\cF$ is a pair $(X,A)$ with $X$ in $\bC$ and $A$ in $F(X)$. A morphism $(X,A)\to (Y,B)$ is a pair $(f,\phi)$ of a morphism
$f:X\to Y$ in $\bC$ and a morphim $\phi:F(f)(A)\to B$ in $F(Y)$.

Assume that the categories $\bC$ and $\cF$ have symmetric monoidal structures such that \begin{equation}\label{werfqelknlweewwefw}
 \pi_{F}((X,A)\otimes_{\cF}(X^{\prime},A^{\prime}))=X\otimes_{\bC}X^{\prime}\ ,  \end{equation} i.e., $\pi_{F}$ preserves the tensor product strictly.
Then, we can write $$(X,A)\otimes_{\cF}(X^{\prime},A^{\prime})=(X\otimes_{\bC} X^{\prime}, A\boxtimes_{X,X^{\prime}}A^{\prime})\ .$$
 For every  two objects $X,X^{\prime}$ in $\bC$ we obtain a bifunctor
\begin{equation}\label{f3foi34jfio34jf34f34f3f}
\boxtimes_{X,X^{\prime}}:F(X)\times F(X^{\prime})\to F(X\otimes_{\bC}X^{\prime})
\end{equation}
which is defined on morphisms in the canonical way.
Let $$f:X\to X^{\prime}\ , \quad g:Y \to Y^{\prime}$$ be morphisms in $\bC$ and $A$ in $F(X)$ and $B$ in $F(Y)$. Then
$$(f,\id_{F(f)(A)}):(X,A)\to (X^{\prime} ,F(f)(A))\ , \quad (g,\id_{F(g)(B)}):(Y,B)\to (Y^{\prime}, F(g)(B))$$  are  morphisms in $\cF$.
Then the second component of their tensor product
$$(f,\id_{F(f)(A)})\otimes_{\cF} (g,\id_{F(g)(B)}) :(X\otimes_{\bC} Y,A\boxtimes_{X,Y}B)\to
(X^{\prime}\otimes_{\bC} Y^{\prime}, F(f)(A) \boxtimes_{X^{\prime},Y^{\prime}}F(g)(B )) $$
 is a morphism \begin{equation}\label{vwrkjbwwejkvjbnkwevewvw}
F(f\otimes_{\bC}g)(A\boxtimes_{X,Y}B)\to  F(f)(A) \boxtimes_{X^{\prime},Y^{\prime}}F(g)(B )
  \end{equation}
  in $F(X^{\prime}\otimes_{\bC}Y^{\prime})$.
  This morphism will appear in the assumptions of the two theorems below.

We now consider the following data:
\begin{enumerate}
\item {a symmetric monoidal $1$-category $\bC$,}
\item {a} functor $F:\bC\to {\mathbf{Cat}}_{1}$
\item {a} symmetric monoidal structure   on the Grothendieck construction $\cF$ of $F$.
\end{enumerate}
Let $\pi_{F}:\cF\to \bC$ denote the associated projection.

\begin{theorem}\label{gui34tefrgeg}
 Assume:
 \begin{enumerate}
\item \label{regoijrogergergf2f} The functor $\pi_{F}$ {strictly} preserves   the tensor product,   the tensor unit as well as the associator, unit, and symmetry transformations.
\item\label{f2f3f23e23e}  \label{regoijrogergergf2f2} For every two  objects $(X,A)$ and $(Y,B)$ in $\cF$ and morphisms $f:X\to X^{\prime}$ and $g:Y\to Y^{\prime}$ in $\bC$ the   morphism
\eqref{vwrkjbwwejkvjbnkwevewvw}
 $$F(f\otimes_{\bC}g)(A\boxtimes_{X,Y} B) \to  F(f)(A)\boxtimes_{X^{\prime}{,}Y^{\prime}} F(g)(B)\ .$$
is an isomorphism.
\end{enumerate}
Then the data provide   a lax symmetric monoidal refinement  (Def. \ref{gfuzfkuzfkuzf})
$$F^{\otimes}:\Nerve(\bC^{\otimes})\to {\mathbf{Cat}}^{\otimes}_{\infty}$$
of the functor $F$.
 \end{theorem}
Note that Condition \ref{regoijrogergergf2f} in the theorem implies the Relation \eqref{werfqelknlweewwefw} so that the bifunctors $\boxtimes_{X,Y}$
appearing in Condition \ref{regoijrogergergf2f2} are, in fact, defined.

The {analoguous} version for additive categories is the following.

Consider the following data:
\begin{enumerate}
\item {a symmetric monoidal $1$-category $\bC$,}
\item {a} functor $F:\bC\to {\mathtt{Add}}_{1}$
\item {a} symmetric monoidal structure   on the Grothendieck construction $\cF$ of $F$.
\end{enumerate}
Let $\pi_{F}:\cF\to \bC$ denote the {associated} projection.

\begin{theorem}\label{rgio34t34r34rerg}
 Assume:
 \begin{enumerate}
\item The functor $\pi_{F}$  {strictly} preserves   the tensor product,   the tensor unit as well as the associator, unit, and symmetry transformations.
\item\label{gioreg34teergegegegeg} The functors $\boxtimes_{X,X^{\prime}}$ are bi-additive for every $X,X^{\prime}$ in $\bC$.
\item  For every two objects $(X,A)$ and $(Y,B)$ in $\cF$ and morphisms $f:X\to X^{\prime}$ and $g:Y\to Y^{\prime}$ in $\bC$ the   morphism
\eqref{vwrkjbwwejkvjbnkwevewvw}
 $$F(f\otimes_{\bC}g)(A\boxtimes_{X,Y} B) \to  F(f)(A)\boxtimes_{X^{\prime},Y^{\prime}} F(g)(B)\ .$$
is an isomorphism.
\end{enumerate}
Then the data provide a lax symmetric monoidal refinement
$$F^{\otimes}:\Nerve(\bC^{\otimes})\to {\mathtt{Add}}^{\otimes}_{\infty}$$
of the functor $F$.

\end{theorem}

\subsection{Proofs of Theorem \ref{gui34tefrgeg} and Theorem \ref{rgio34t34r34rerg}.}

We start with the proof of Theorem \ref{gui34tefrgeg}.
Let  $$\Nerve(\bC^{\otimes})\to {\Nerve}(\mathbf{Fin}_{*})$$ denote the symmetric monoidal $\infty$-category  corresponding to
the symmetric monoidal category $\bC$ \cite[Ex. 2.1.2.21]{HA}. Let $${\mathbf{Cat}}^{\otimes}_{\infty}\to   {\Nerve}(\mathbf{Fin}_{*})$$ be the     cocartesian fibration corresponding to
the symmetric monoidal category of small categories.
Then the $\infty$-category $${\mathbf{Alg}}_{\Nerve(\bC)}({\mathbf{Cat}}_{\infty}):=\mbox{\{operad maps $\Nerve(\bC^{\otimes})\to {\mathbf{Cat}}_{\infty}^{\otimes}$\}}$$ corresponds to the $\infty$-category of lax symmetric monoidal functors
$$\Nerve(\bC)\to {\mathbf{Cat}}_{\infty}\ ,$$
see the text after \cite[Rem. 2.1.3.6]{HA}.
We let $$\Mon_{\Nerve(\bC)}({\mathbf{Cat}}_{\infty})$$ denote the category of $\Nerve(\bC)$-monoids in ${\mathbf{Cat}}_{\infty}$
 \cite[Def. 2.4.2.1]{HA}. By  \cite[Prop. 2.4.2.5]{HA}, we have an equivalence
 $${\mathbf{Alg}}_{\Nerve(\bC)}({\mathbf{Cat}}_{\infty})\simeq \Mon_{\Nerve(\bC)}({\mathbf{Cat}}_{\infty})\ .$$
By  \cite[Rem. 2.4.2.4]{HA}, in order to provide an object of  $\Mon_{\Nerve(\bC)}({\mathbf{Cat}}_{\infty})$, it suffices to
present a cocartesian fibration $$p:\cC^{\otimes}\to \bC^{\otimes}$$ which exhibits
$\cC^{\otimes}$ as a   $\Nerve(\bC)$-monoidal category \cite[Rem. 2.1.2.13]{HA}.
To this end we must show that the composition
$$\cC^{\otimes}\to \bC^{\otimes}\to {\Nerve}(\mathbf{Fin}_{*})$$
exhibits $\cC^{\otimes}$ as an $\infty$-operad \cite[Prop. 2.1.2.12]{HA}.

Let $$\pi_{F}:\cF\to \bC$$ be a symmetric monoidal functor between $1$-categories as in the Theorem  \ref{gui34tefrgeg}.
We get
an induced functor of symmetric monoidal categories $$\pi_{F}^{\otimes}:\cF^{\otimes}\to \bC^{\otimes}$$ and {thus} a morphism of $\infty$-operads
 $$\Nerve(\pi^{\otimes}_{F}):\Nerve(\cF^{\otimes})\to \Nerve(\bC^{\otimes})\ .$$ Our task is then to show that
 $\Nerve(\pi_{F}^{\otimes})$ exhibits
$\Nerve(\cF^{\otimes})$ as an $\Nerve(\bC)$-monoidal category, and we must only check that
$\Nerve(\pi_{F}^{\otimes})$ is a cocartesian fibration.
It suffices to check that $\pi_{F}^{\otimes}$ is an op-fibration of $1$-categories.

By assumption,  the underlying functor of   $\pi_{F} $ (after forgetting the symmetric monoidal structures)   arose from a Grothendieck construction for a functor $$F:\bC\to {\mathbf{Cat}}_{1}\ .$$
Recall from \cite[Constr. 2.0.0.1]{HA} that the objects of $\bC^{\otimes}$ in the fibre  $\bC\langle n\rangle $ of $\bC^{\otimes}$ over $\langle n\rangle $ in $\mathbf{Fin}_{*}$ are $n$-tuples of objects of $\bC$.
Consider  two objects $$(X_{1},\dots,X_{n})\ , \quad (Y_{1},\dots,Y_{m})$$ in $\bC^{\otimes}\langle n\rangle $ and $\bC^{\otimes}\langle m\rangle $ and an object $$((X_{1},A_{1}),\dots,(X_{n},A_{n}))$$ in
$\cF^{\otimes}\langle n\rangle $, where $A_{i}$ belongs to $F(X_{i})$. Let $\alpha\langle n\rangle \to \langle m\rangle $ be a morphism in $\mathbf{Fin}_{*}$ and $$f:
(X_{1},\dots,X_{n})\to (Y_{1},\dots,Y_{m})$$  be a morphism in $\bC^{\otimes}$ over $\alpha$. Then $f$ is given by   a collection of morphisms
$ f:=(f_{j})_{j\in \langle m\rangle }$  with $$f_{j}:\otimes_{i\in \alpha^{-1}(j)} X_{i}\to Y_{j}\ .$$  We must provide a cocartesian lift of $f$.  For $j$ in $\langle m\rangle$
we have a morphism
$$g_{j}{:=}(f_{j},\id_{\boxtimes_{i\in \alpha^{-1}(j)}A_{i}}):(\otimes_{i\in \alpha^{-1}(j)} X_{i},\boxtimes_{i\in \alpha^{-1}(j)}A_{i})\to (Y_{j},F(f_{j})(\boxtimes_{i\in \alpha^{-1}(j)}A_{i}))$$ in $\cF$.
 One now checks in a straightforward (but tedious) manner that the collection
$g:=(g_{j})_{j\in \langle m\rangle }$ is the cocartesian lift of $f$.
The argument repeatedly uses the Condition \ref{f2f3f23e23e}. This finishes the proof of Theorem \ref{gui34tefrgeg}.

 We now turn to the proof of Theorem \ref{rgio34t34r34rerg}.
Consider the symmetric monoidal subcategory
$${\mathtt{Add}}_{\infty}^{\otimes}\subseteq {\mathbf{Cat}}_{\infty}^{\otimes}\ .$$

Indeed we can {first} consider the subcategory ${\mathbf{Cat}}_{\infty}(\coprod)$ of $\infty$-categories {which} admit finite coproducts and coproduct preserving functors.
By  \cite[Cor. 4.8.1.4]{HA} (applied to  the collection  {$\mathcal{K}$} of finite sets) we get a symmetric monoidal subcategory
$${\mathbf{Cat}}_{\infty}(\coprod)^{\otimes}\to  {\mathbf{Cat}}_{\infty}^{\otimes}\ .$$
In the next step we view
${\mathtt{Add}}_{\infty}$ as a full subcategory of ${\mathbf{Cat}}_{\infty}(\coprod) $ of pointed $1$-categories in which products and coproducts coincide. Using  \cite[Cor. 2.2.1.1]{HA} one then shows that
$${\mathtt{Add}}_{\infty}^{\otimes}\to  {\mathbf{Cat}}_{\infty}(\coprod)^{\otimes}$$
is again a suboperad.
We now consider the diagram
$$\xymatrix{{\mathtt{Add}}_{\infty}^{\otimes}\ar[r]& {\mathbf{Cat}}_{\infty}(\coprod)^{\otimes}\ar[d]\\\Nerve(\bC^{\otimes})\ar[r]\ar@{-->}[u]\ar@{..>}[ur]&{\mathbf{Cat}}_{\infty}^{\otimes}}\ .$$
The lower horizontal map is a morphism of {$\infty$-}operads by Theorem \ref{gui34tefrgeg}.
We first argue that the dotted lift exists. To this end we use  \cite[Notation 4.8.1.2]{HA}. One must check that
 $F$ takes values in categories admitting finite coproducts (clear), and that the functors
$$\boxtimes_{X,Y}:F(X)\times F(Y)\to F(X\times Y)$$ preserves
sums in both variables separately, i.e., Assumption \ref{gioreg34teergegegegeg}.
Finally,  for the dashed arrow we use that
$F$ takes values in ${\mathtt{Add}}_{{1}}$.

\section{The symmetric monoidal functor of controlled objects}

\subsection{Symmetric monoidal structures}\label{ewfoiewfwefwefwef}

In this subsection we write out, for later reference, the structures of a symmetric monoidal category and of a (lax) symmetric monoidal functor.
Let $\bC$ be a $1$-category: \begin{ddd}\label{gui23r32reger} {\cite[Sec. VII. 1. \& 7.]{maclane}}
A symmetric monoidal structure on $\bC$ is given by the following data:
\begin{enumerate}
\item\label{gui23r32reger1} a bifunctor $(-\otimes_{\bC}-):\bC\times \bC\to \bC$,
\item\label{gui23r32reger2} an object $1_{\bC}$ (the tensor unit),
\item a natural isomorphism  (the associativity constraint) $$\alpha^{\bC}:(-\otimes_{\bC}-)\circ ((-\otimes_{\bC}-) \times  \id_{\bC} )  \to (-\otimes_{\bC}-)\circ  (\id_{\bC}\times (-\otimes_{\bC}-))   \ ,$$
\item a  natural isomorphism $\eta^{\bC}:1_{\bC}\otimes_{\bC}-\to \id_{\bC}$ (the unit constraint),
\item a  natural isomorphism (the symmetry) $\sigma^{\bC}:(-\otimes_{\bC}-)\circ T\to (-\otimes_{\bC}-)$, where $T:\bC\times \bC\to \bC\times \bC$ is the flip functor.
\end{enumerate}
This data have to satisfy the following relations:
\begin{enumerate}
\item the pentagon relation,
\item  the triangle relation,
\item the inverse relation,
\item the associativity coherence.
\end{enumerate}
\end{ddd}
A symmetric monoidal category is a category {equipped} with a symmetric monoidal structure.

We will use the name of the category as a superscript for the constraints, but if we evaluate e.g. the symmetry constraint $\sigma^{\bC}$ at the objects $C,C^{\prime}$ of $\bC$, then we write shortly $\sigma_{C,C^{\prime}}$ instead of $\sigma^{\bC}_{C,C^{\prime}}$ since the type of objects in the subscript already determines the category in question.

Let $\bC$ and $\bD$ be   symmetric monoidal categories,  and let $F:\bC\to \bD$ be a functor.
\begin{ddd}\label{34f8924fregerg} {\cite[Sec. XI. 2.]{maclane}}
A symmetric monoidal structure on $F$ is given by the following data:
\begin{enumerate}
\item\label{3gio34t3434g} an isomorphism  $\epsilon^{F}:1_{\bD}\to F(1_{\bC})$,
\item\label{3gio34t3434g1} a natural isomorphism $\mu^{F}:(- \otimes_{\bD}-)\circ (F\times F)\to F\circ ( -\otimes_{\bC}-)$.
\end{enumerate}
This data have to satisfy the following relations:
\begin{enumerate}
\item associativity relation,

\item\label{fiowfewef23rr3} unitality relation,
\item\label{evfiwehfioewfewfewfewff} symmetry relation.
\end{enumerate}

\end{ddd}

\begin{rem}\label{gir2i23joijoir23r23r}
If we weaken the {assumptions} and {we} only require   that $\epsilon^{F}$ and $\mu^{F}$ are
natural transformations, then we get the definition of a lax symmetric monoidal functor.
\end{rem}

\subsection{Bornological coarse spaces }\label{geriogjheroigfwefwefw}

In this subsection we recall the definition {of} the symmetric monoidal category $G\mathbf{BornCoarse}$ of $G$-bornological coarse spaces \cite[Sec. 2]{buen}, \cite[Sec. 2.1]{equicoarse}.

In the  definitions below we will use the following {notation:}
\begin{enumerate}
\item For a set $Z$ we let $\mathcal{P}(Z)$ denote the power set of $Z$.
\item If  a group $G$ acts on a set $X$, then it acts diagonally on $X\times X$ and therefore on $\mathcal{P}(X\times X)$.
For $U$ in $\mathcal{P}(X\times X)$ we set \begin{equation*}\label{betrboklvree}
GU:=\bigcup_{g\in G} gU\ .
\end{equation*}
\item For $U$ in $\mathcal{P}(X\times X)$ and $B$ in $\mathcal{P}(X)$ we define the
 $U$-thickening $U[B]$   by
\begin{equation}\label{viojoewfwefvvdvsdv} U[B]:=\{x\in X \mid \exists y\in B: (x,y) \in U \}\ .\end{equation}
\item For $U$ in $\mathcal{P}(X\times X)$ we define {its} inverse  by $$U^{-1}:=\{(y,x) \mid (x,y)\in U  \}\ .$$  \item For $U,V$ in $\mathcal{P}(X\times X)$ we define their composition  by \begin{equation}\label{rv3roih43iuoff3fwe}
U\circ V:=\{(x,z) \mid \exists y\in X : (x,y)\in U \wedge (y,z)\in V  \}\ .
\end{equation}

\end{enumerate}

Let $G$ be a group and let $X$ be a $G$-set.

\begin{ddd}
 A {$G$-coarse structure} $\cC$ on $X$ is a subset of $\mathcal{P}(X\times X)$ with the following properties:
\begin{enumerate}
\item $\cC$ is closed under composition, inversion, and forming finite unions or subsets.
\item $\cC$ contains the diagonal $\diag(X)$ of $X$.
\item {For} every $U$ in $\cC$, the set $GU$
is also in $\cC$.
\end{enumerate}
The pair $(X,\cC)$ is called a  {$G$-coarse space},
 {and the members of $\cC$ are called (coarse)  {entourages} of $X$.}
\end{ddd}

Let $(X,\cC)$ and $(X^{\prime},\cC^{\prime})$ be $G$-coarse spaces and  {let} $f\colon X\to X^{\prime}$ be an equivariant map between the underlying sets.
\begin{ddd}
The map $f$ is  {controlled} if for every $U$ in $\cC$ we have  $(f\times f)(U)\in \cC^{\prime}$.  \end{ddd}

We obtain a category $G\mathbf{Coarse}$ of $G$-coarse spaces and controlled equivariant maps.

Let $G$ be a group and let $X$ be a $G$-set.
\begin{ddd}\label{erjgoijgoiergjergegrrr}A  {$G$-bornology} $\cB$ on  $X$ is a subset of $\mathcal{P}(X)$ with the following properties:
\begin{enumerate}
\item $\cB$ is closed under {forming} finite unions and subsets.
\item $\cB$ contains all finite subsets of $X$.
\item $\cB$ is $G$-invariant.
\end{enumerate}
The pair $(X,\cB)$ is called a  {$G$-bornological  space}{, and the members of $\cB$ are called  {bounded subsets} of $X$.}
\end{ddd}

Let $(X, {\cB})$ and $(X^{\prime}, {\cB^{\prime}})$ be $G$-bornological spaces and {let} $f \colon X\to X^{\prime}$ be an equivariant map between the underlying sets.

\begin{ddd}\label{erjgoijgoiergjergegrrr1}
 {The map} $f$ is {proper} if for every $B^{\prime}$ in $\cB^{\prime}$ we have $f^{-1}(B^{\prime})\in  \cB$.   \end{ddd}

We obtain a category $G\mathbf{Born}$ of $G$-bornological spaces and proper equivariant maps.

Let $X$ be a $G$-set {equipped} with a $G$-coarse structure $\cC$ and a $G$-bornology $\cB$.
\begin{ddd}
The coarse structure $\cC$ and the bornology $\cB$ are said to be
  {compatible} if
for every $B$ in $\cB$ and $U$ in $\cC$
 {the $U$-thickening $U[B]$ (see \eqref{viojoewfwefvvdvsdv}) lies in $\cB$.}
\end{ddd}

\begin{ddd}\label{ergergregreg5t45t}
A  {$G$-bornological coarse space} is a triple $(X,\cC,\cB)$ consisting of a $G$-set $X$, a $G$-coarse structure $\cC$ and a $G$-bornology $\cB$ {on $X$,} such that $\cC$ and $\cB$  are  compatible. \end{ddd}

Usually we will denote a $G$-bornological coarse space by the symbol $X$ and write $\cB(X)$ and $\cC(X)$ for its bornology and coarse structures.

\begin{ddd}
A  {morphism} $f:X \to X^{\prime} $ between $G$-bornological coarse spaces
is an equivariant  map  of the {underlying} $G$-sets {that} is controlled and proper.
\end{ddd}

We obtain a category $G\mathbf{BornCoarse}$ of $G$-bornological coarse spaces and morphisms.

Next we describe the symmetric monoidal structure on $G\mathbf{BornCoarse}$ \cite[Ex. 2.17]{equicoarse}. We have a forgetful functor
$$U:G\mathbf{BornCoarse}\to G\Set$$ which associates to every $G$-bornological coarse space $X$ {its} underlying $G$-set.
This functor is faithful.
The category $G\Set$ {is endowed with} the cartesian symmetric monoidal structure.  The symmetric monoidal structure on $G\mathbf{BornCoarse}$ will be defined {in such a way}  that the functor $U$ preserves the unit and the tensor product strictly, i.e., the morphisms  \ref{3gio34t3434g} and \ref{3gio34t3434g1} in Definition \ref{34f8924fregerg} are identities. In other words, the associator, unit and symmetry constraints are imported from $G\Set$ and
satisfy the relations required in Definition \ref{gui23r32reger} automatically.

We start with the description of the bifunctor
$$-\otimes_{G\mathbf{BornCoarse}}-:G\mathbf{BornCoarse}\times G\mathbf{BornCoarse}\to G\mathbf{BornCoarse}\ .$$
Let $X $ and  $ X^{\prime} $ be two $G$-bornological coarse spaces. Then
their tensor product $$ X\otimes_{G\mathbf{BornCoarse}}  X^{\prime} $$  is the $G$-bornological coarse spaces defined as follows:
\begin{enumerate}
\item The underlying $G$-set of $X\otimes_{G\mathbf{BornCoarse}}  X^{\prime}$ is the cartesian product of the underlying $G$-sets $X\times X^{\prime}$.
\item The $G$-bornology on $X\times X^{\prime}$ is generated by the subsets $B\times B^{\prime}$ for all $B$ in $\cB(X)$ and $B^{\prime}$ in $\cB(X^{\prime})$.
\item The $G$-coarse structure on $X\times X^{\prime}$ is generated by the entourages $U\times U^{\prime}$ for $U$ in $\cC(X)$ and $U^{\prime}$ in $\cC(X^{\prime})$.
\end{enumerate}
Here a $G$-bornological (or coarse, respectivley) structure generated by a family of   subsets (or entourages)  is the minimal $G$-bornological (or $G$-coarse) structure containing these subsets (or entourages).
Note that the underlying $G$-coarse space of the tensor product  represents the cartesian product of the underlying $G$-coarse spaces of the factors in $G\mathbf{Coarse}$, but the tensor product is not the cartesian product in $G\mathbf{BornCoarse}$ in general.

From now on we will use the shorter notation $X\otimes X^{\prime}$ for the tensor product {of $G$-bornological coarse spaces}, i.e., we omit the subscript {$G\mathbf{BornCoarse}$.}

If $f:X\to Y$ and $f^{\prime}:X^{\prime}\to Y^{\prime}$ are morphisms of $G$-bornological coarse spaces, then their tensor product
$$f\otimes f^{\prime}:X\otimes Y\to X^{\prime}\otimes Y^{\prime}$$ is induced by {the}  equivariant map of underlying $G$-sets $(x,y)\mapsto (f(x),f(y))$.
This finishes the description of the bifunctor \ref{gui23r32reger}.\ref{gui23r32reger1}

The tensor unit $1_{G\mathbf{BornCoarse}}$ (\ref{gui23r32reger}.\ref{gui23r32reger2})   is given  by the one-point space $*$.

As explained above, the associativity, unit and symmetry constraints are imported from $G\Set$. It is straightforward to check that they are implemented by  morphisms of $G$-bornological coarse spaces.

This finishes the description of the symmetric monoidal structure $\otimes$ on the category  $G\mathbf{BornCoarse}$.

\subsection{Controlled objects}

In this section, for every additive category ${\mathbf{A}}$ with a strict $G$-action, we describe the functor $$\bV_{{\mathbf{A}}}^{G}:G\mathbf{BornCoarse}\to {\mathtt{Add}}_{1}$$ which sends a $G$-bornological coarse space $X$ to its additive category $\bV_{{\mathbf{A}}}^{G}{(X)}$ of equivariant $X$-controlled ${\mathbf{A}}$-object \cite[Sec.~8.2]{equicoarse}.

{For a group $G$, l}et $BG$ be the category with one object $*$ and $\End_{BG}(*)\cong G$. Then  $\Fun(BG,{\mathtt{Add}}_{1})$ is the category of additive categories with {a} strict $G$-action. Explicitly, an additive category with a strict $G$-action is an additive category ${\mathbf{A}}$ (the evaluation of the functor at the object $*$ in $BG$)  together with an action of $G$ on ${\mathbf{A}}$ by exact functors, which is strictly associative. Our notation   for the action of $g$ in $G$ on objects $A$ {of ${\mathbf{A}}$}  and morphisms $f$ is
$$(g,A)\mapsto gA\ , \quad (g,f:A\to A^{\prime})\mapsto (gf:gA\to gA^{\prime})\ .$$

  Let ${\mathbf{A}}$ be an additive category with a strict $G$-action and $X$ be a $G$-bornological coarse space. We consider the bornology  $\cB(X)$ of $X$ as a poset with a $G$-action $(g,B)\mapsto gB$,  hence as a category with a strict $G$-action, i.e., an object of  $\Fun(BG,{\mathbf{Cat}}_{1})$.

The category $\Fun(\cB(X),{\mathbf{A}})$ has an induced $G$-action which can explicitly be described as follows. If $M:\cB\to {\mathbf{A}}$ is a functor and $g$ is an element of $G$, then  $g M\colon \cB(X)\to {\mathbf{A}}$ is  the functor which sends a bounded set $B$ in $\cB(X)$ to the object $gM(g^{-1}(B))$ of ${\mathbf{A}}$.
If $\rho:M\to  M^{\prime}$ is a natural transformation between two such functors, then we let  $g\rho :gM\to gM^{\prime}$  denote the canonically induced natural transformation.

\begin{ddd}\label{def:Xcontrolledobject}{\cite[Def. 8.3]{equicoarse}}
	An {equivariant $X$-controlled ${\mathbf{A}}$-object} is a pair $(M,\rho)$ consisting of a functor
	$M \colon \cB(X) \to {\mathbf{A}}$
	and a family $\rho=(\rho(g))_{g\in G}$ of natural isomorphisms
	$\rho(g)\colon M \to g M$
	 	satisfying the following conditions:
	\begin{enumerate}
		\item\label{def:Xcontrolledobject:it1} $M(\emptyset) \cong 0$.
		\item\label{def:Xcontrolledobject:it3} For all $B, B'$ in $\cB(X)$, the commutative square
		\[\xymatrix{
			M(B \cap B')\ar[r]\ar[d] & M(B)\ar[d] \\
			M(B')\ar[r] & M(B \cup B')
		}\]
		is a pushout square.
		\item\label{def:Xcontrolledobject:it4} For all $B$ in $\cB(X)$ there exists a finite subset $F$ of $B$ such that the inclusion $F\to B$ induces an isomorphism   $M(F) \xrightarrow{\cong} M(B)$. 		\item \label{def:Xcontrolledobject:it5}For all pairs of elements  $g,g'$ of $G$ we have the relation
		$\rho(g g^{\prime})= g\rho(g')\circ \rho(g)$.
		\qedhere
	\end{enumerate}
\end{ddd}

If $U$ is an invariant coarse entourage of $X$, i.e.,  an element of $\cC(X)^{G}$, then we get a $G$-equivariant functor
$$U[-]:\cB(X)\to \cB(X)$$ which sends a bounded
 subset $B$ of $X$ to its  $U$-thickening   $U[B]$, see \eqref{viojoewfwefvvdvsdv}. Indeed,  the $U$-thickening   $U[B]$ of a bounded subset $B$    is again bounded by the compatibility of the coarse structure $\cC(X)$ and the bornology $\cB(X)$ of $X$, and $U[-]$ preserves the inclusion relation. Since  $U$  is $G$-invariant we have   the equality  $U[gB]=gU[B]$. It implies that $U[-]$  is $G$-{equivariant}.
 If $M:\cB(X)\to {\mathbf{A}}$ is a functor, then we write $U[-]^{*}M:=M\circ U[-]$ for the pull-back of $M$ along $U[-]$.

Let $(M,\rho), (M',\rho')$ be two equivariant $X$-controlled ${\mathbf{A}}$-objects and $U$ be  an invariant coarse entourage of $X$.
\begin{ddd}\label{rebgieorgergergegerg}
	An {equivariant $U$-controlled morphism} $\phi  \colon (M,\rho) \to (M',\rho')$ is a natural transformation
	\[ \phi \colon M  \to U[-]^{*}M' \ ,\]
	such that $\rho'(g)\circ \phi =(g \phi )\circ \rho(g)$ for all elements $g$ of $G$.
\end{ddd}
We let ${\mathtt{Mor}}_U((M,\rho),(M',\rho'))$ denote the abelian group of equivariant $U$-controlled morphisms.

If $U^{\prime}$ is in $\cC(X)^{G}$ such that $U\subseteq U^{\prime}$, then for every $B$ in $\cB(X)$ we have $U[B]\subseteq U^{\prime}[B]$. These inclusions induce a transformation between functors
$U[-]^{*}M^{\prime} \to U^{\prime}[-]^{*} M^{\prime} $ and therefore a map
$${\mathtt{Mor}}_U((M,\rho),(M',\rho'))\to {\mathtt{Mor}}_{U^{\prime}}((M ,\rho),(M^{\prime},\rho'))$$
by postcomposition. Using these maps in the interpretation of the colimit
 we define the abelian group of equivariant controlled morphisms from $(M,\rho)$ to $(M',\rho^{\prime})$ by
\[ \Hom_{\bV^{G}_{\mathbf{A}}(X)}((M,\rho),(M',\rho')) := \colim_{U \in \cC(X)^{G}} {\mathtt{Mor}}_U((M,\rho),(M',\rho'))\ .\]

We now consider a pair  of morphisms in $$ \Hom_{\bV^{G}_{\mathbf{A}}(X)}((M,\rho),(M',\rho'))\:\:\mbox{and}\:\:  \Hom_{\bV^{G}_{\mathbf{A}}(X)}((M^{\prime},\rho^{\prime}),(M^{\prime\prime},\rho^{\prime\prime}))\ ,$$ respectively,
 which are represented by
  $$\phi :M \to U[-]^{*}M^{\prime} \:\:\mbox{and}\:\: \phi ^{\prime}:M^{\prime} \to U^{\prime}[-]^{*}M^{\prime\prime} \ .$$
We define the composition of the two morphisms  to be represented by the morphism
$$U[-]^{*}\phi ^{\prime}\circ \phi : M\to (U^{\prime}\circ U)[-]^{*}M^{\prime\prime} \ , $$ (see \eqref{rv3roih43iuoff3fwe} for notation)
where
$$U[-]^{*}\phi^{\prime}:U[-]^{*}M^{\prime} \to (U^{\prime}\circ U)[-]^{*}M^{\prime\prime} $$ is defined in the canonical manner.
We denote the resulting   category of equivariant $X$-controlled ${\mathbf{A}}$-objects and equivariant controlled morphisms by $\bV_{\mathbf{A}}^G(X)$. This category is additive {\cite[Lemma 8.7]{equicoarse}}.

Let $f \colon X  \to X'  $  be a morphism of $G$-bornological coarse spaces,
and let $(M,\rho)$ be an equivariant $X$-controlled ${\mathbf{A}}$-object.
Since $f$ is proper, it induces an equivariant functor $f^{-1}:\cB(X^{\prime})\to \cB(X)$, and we can define a functor $f_*M \colon \cB(X^{\prime}) \to {\mathbf{A}}$ by
\[ f_*M := M\circ f^{-1} \ .\]
Furthermore, we define
\[f_*\rho(g) :=\rho(g)\circ f^{-1} \ .\]
Let $U$ be in $\cC(X)^{G}$ and let $\phi \colon (M,\rho) \to (M',\rho')$ be an equivariant $U$-controlled morphism. Then  $V:= (f \times f)(U)$ belongs to $\cC(X^{\prime})^{ G}$ and we have  $U[f^{-1}(B^{\prime})] \subseteq f^{-1}(V[B^{\prime}])$ for all bounded subsets $B^{\prime}$ of $X^{\prime}$. Therefore, we obtain an induced $V$-controlled morphism
\[ f_*\phi = \{  f_*M(B^{\prime})\stackrel{\phi_{f^{-1}(B^{\prime})}}{ \to} M(U[f^{-1}(B^{\prime})])\to  f_*M(V[B^{\prime}]) \}_{B^{\prime} \in \cB(X')}\ .\]
One checks that this construction defines an additive functor
\[ f_* \colon \bV^G_{\mathbf{A}}(X) \to \bV^G_{\mathbf{A}}(X')\ .\]
This completes the construction of the functor  \begin{equation}\label{ekrjekrferferferf}
\bV^G_{\mathbf{A}} \colon G\mathbf{BornCoarse} \to {\mathtt{Add}}_{{1}}\ .
\end{equation}

In the following we give a more explicit description of the objects and morphisms in $\bV_{{\mathbf{A}}}^{G}(X)$ which will be used in the description of the  symmetric monoidal structure on the Grothendieck construction {associated to the functor} $\bV_{{\mathbf{A}}}^{G}$ in Section \ref{vevgebvveerverv}.

\begin{conv}\label{choosesum}
	We consider an additive category  ${\mathbf{A}}$. If $(A_{i})_{i\in I}$ is a  family of objects of ${\mathbf{A}}$ with at most finitely many non-zero members, then we use the symbol
	$\bigoplus_{i\in I} A_{i}$ in order to
	denote a choice of an object of ${\mathbf{A}}$  {together}  with a family of morphisms $(A_{j}\to  \bigoplus_{i\in I} A_{i})_{j\in I}$ {representing} the coproduct of the family.

Since in an additive category coproducts and products coincide, for every $j$ in $I$ we  furthermore have a   canonical projection   $$ \bigoplus_{i\in I} A_{i} \to A_{j}$$
such that the diagram
 $$\xymatrix{A_{j}\ar[r]\ar@/^0.7cm/[rr]^{{\id_{A_{j}}}}&\bigoplus_{i\in I} A_{i} \ar[r]&A_{j}}$$
 commutes.

If $(A_{i^{\prime}}^{\prime})_{i^{\prime}\in I^{\prime}}$ is a second   family of this type
and $(\phi_{i,i^{\prime} }:A_{i^{\prime}}^{\prime}\to A_{i})_{i^{\prime}\in I^{\prime},i\in I}$ is a family of morphisms in ${\mathbf{A}}$, then we have a
unique  morphism $\oplus \phi_{i,i^{\prime}}$   such that the squares \begin{equation}\label{vejebvierhviervervev}
\xymatrix{A_{i^{\prime}}^{\prime}\ar[r]^{\phi_{i,i^{\prime}}}\ar[d]&A_{i}\ar[d]\\\bigoplus_{i^{\prime}\in I^{\prime}} A_{i^{\prime}}^{\prime} \ar[r]^{\oplus \phi_{i,i^{\prime}}}&\bigoplus_{i\in I} A_{i}}
\end{equation}
commute for every $i^{\prime}$ in $I^{\prime}$ and $i$ in $I$.\hspace*{\fill}$\qed$}% \newline\noindent
\end{conv}

{Let ${\mathbf{A}}$ be a small additive category with strict $G$-action.}
Let $X  $ be a  $G$-bornological coarse space  (see Definition \ref{ergergregreg5t45t}),  and let $(M,\rho)$ be an equivariant $X$-controlled ${\mathbf{A}}$-object (see Definition \ref{def:Xcontrolledobject}). Let $B$ be  in $\cB(X)$ and $x$ be a point in $B$. The inclusion $\{x\}\to B$ induces a morphism
$M(\{x\})\to M(B)$ in ${\mathbf{A}}$. The
conditions \ref{def:Xcontrolledobject}.\ref{def:Xcontrolledobject:it1} and \ref{def:Xcontrolledobject}.\ref{def:Xcontrolledobject:it3}
together imply that
$M(\{x\})=0$ for all but finitely many points of $B$, and that the canonical morphism (induced by the universal property of the coproduct in ${\mathbf{A}}$)
\begin{equation}\label{g5giojgoigg5g45gfg}
\bigoplus_{x\in B} M(\{x\}) \xrightarrow{\cong} M(B)\
\end{equation}
{is an isomorphism.}

Let now $U$ be in $\cC(X)^{G}$, and let $\phi:(M,\rho)\to (M^{\prime},\rho^{\prime})$ be an equivariant  $U$-controlled morphism. By Definition \ref{rebgieorgergergegerg}, {the morphism $\phi$} is given by   a natural transformation of functors $\phi:M\to U[-]^{*}M^{\prime}$ satisfying an equivariance condition.
For every point $x$ in $X$ we get a morphism
\begin{equation}\label{dewd23d2d3d2}
M(\{x\})\to M^{\prime}(U[\{x\}])\stackrel{\eqref{g5giojgoigg5g45gfg}}{\cong}
\bigoplus_{x^{\prime}\in U[\{x\}]} M(\{x^{\prime}\})\
\end{equation}
{in ${\mathbf{A}}$.}
We let\begin{equation}\label{vrtvoijogi43ferv}
\phi_{x^{\prime},x}:M(\{x\})\to M^{\prime}(\{x^{\prime}\})
\end{equation}   denote the composition of  {\eqref{dewd23d2d3d2}} with the projection onto the summand corresponding to $x^{\prime}$.
In this way we get a family of morphisms $(\phi_{x^{\prime},x})_{x^{\prime},x\in X}$ in ${\mathbf{A}}$. In a similar manner, for $g$ in $G$, the transformation $\rho(g):M\to gM$ gives rise to a family of morphisms
\begin{equation}\label{v4rr4vhu34fr3f}
\big(\rho(g)_{x}:M(\{x\})\to gM(\{g^{-1}x\})\big)_{x\in X}\ .
\end{equation}
By construction the family    $(\phi_{x^{\prime},x})_{x^{\prime},x\in X}$ satisfies the following conditions.
\begin{enumerate}

	\item\label{efghewufw23} For all $x,x^{\prime}$ in $X$ the condition
	$\phi_{x^{\prime},x}\not=0$ implies that  $(x^{\prime},x )\in U$.  \item\label{efghewufw232} We have
	$\rho^{\prime}(g)_{x^{\prime}}\circ \phi_{x^{\prime},x}=
	(g\phi)_{g^{-1}x^{\prime},g^{-1}x}\circ \rho(g)_{x}$
	for all $x,x^{\prime}$ in $X$ and $g$ in $G$.
\end{enumerate}

\begin{lem}\label{fiofhoiwefhwef32r} We have  a bijection between equivariant $U$-controlled morphisms $\phi:(M,\rho)\to (M^{\prime},\rho^{\prime})$
	and families
	$(\phi_{x^{\prime},x})_{x^{\prime},x\in X}$ of morphisms as in \eqref{vrtvoijogi43ferv} satisfying the {C}onditions  \ref{efghewufw23} and \ref{efghewufw232}.

	%

\end{lem}

\begin{proof}
Let $(M,\rho)$  and $(M^{\prime},\rho^{\prime})$ be in $\bV_{{\mathbf{A}}}^{G}(X)$.
We must show that a matrix $(\phi_{x^{\prime},x})_{x^{\prime},x\in X}$ of morphisms
 as in \eqref{vrtvoijogi43ferv} which satisfies  the {C}onditions  \ref{efghewufw23} and \ref{efghewufw232}
 gives rise to a{n equivariant controlled} morphism $\phi:(M,\rho)\to (M^{\prime},\rho^{\prime})$. Let $U$ be in $\cC^{G}(X)$ such that {C}ondition  \ref{efghewufw23} holds true. We must construct an equivariant  natural transformation
 $\phi:M\to U[-]^{*}M^{\prime}$.

 We consider
 $B$ in $\cB(X)$. Then
 $(M(\{x\})_{x\in B}$ and $(M^{\prime}(\{x^{\prime}\}))_{x^{\prime}\in U[B]}$ are families of objects in ${\mathbf{A}}$ with at most finitely many non-zero members.

 Using Convention \ref{choosesum},  and in particular  the notation from \eqref{vejebvierhviervervev},
  we  can define the morphism
 $\phi_{B}:M(B)\to M^{\prime}(U[B])$ such that
 $$\xymatrix{\bigoplus_{x\in B}M(\{x\})\ar[rr]^{\oplus \phi_{x^{\prime},x}}\ar[d]^{\eqref{g5giojgoigg5g45gfg}}_{\cong}&&\bigoplus_{x^{\prime}\in U[B]}M^{\prime}(\{x^{\prime}\})\ar[d]^{\eqref{g5giojgoigg5g45gfg}}_{\cong}\\M(B)\ar[rr]^{\phi_{B}}&&M^{\prime}(U[B])}$$
 commutes. It is now straightforward to check that the family $(\phi_{B})_{B\in \cB(X)}$ assembles to a natural transformation $\phi:M\to U[-]^{*}M^{\prime}$ as required. By construction the morphism $\phi$ is $U$-controlled.  Furthermore, the Condition \ref{efghewufw232} implies that $\phi$ satisfies the equivariance condition stated in {Definition} \ref{rebgieorgergergegerg}. \end{proof}

Let ${f}:X_{0}\to X_{1}$ be a morphism of $G$-bornological coarse spaces and $(M_{i},\rho_{i})$ be  objects of $\bV_{{\mathbf{A}}}^{G}(X_{i})$ for $i=0,1$.
Then a morphism \begin{equation}\label{f3oifjiocvcwecw}
\phi:f_{*}(M_{0},\rho_{0})\to (M_{1},\rho_{0})
\end{equation}
induces a  matrix \begin{equation}\label{f34flk3n4fok34f34f}
\left(\phi^{f}_{x_{1},x_{0}}:M_{0}(\{x_{0}\})\to M_{1}(\{x_{1}\})\right)_{x_{0}\in X_{0}, x_{1}\in X_{1}} \ .
\end{equation}
To this end we observe that
$$(f_{*}M_{0})(\{x^{\prime}_{1}\})=M_{0}(f^{-1}(\{x_{1}^{\prime}\}))\stackrel{\eqref{g5giojgoigg5g45gfg}}{\cong} \bigoplus_{x_{0}\in f^{-1}(\{x_{1}^{\prime}\})} M_{0}(\{x_{0}\})$$ so that
$$\left(\phi_{x_{1} ,x_{1}^{\prime}}:=\oplus_{x_{0}\in f^{-1}(\{x_{1}^{\prime}\})} \phi^{f}_{x_{0},x_{1}}:M_{0}(f^{-1}(\{x_{1}^{\prime}\}))\cong\bigoplus_{x_{0}\in f^{-1}(\{x_{1}^{\prime}\})} M_{0}(\{x_{0}\})
 \to M_{1}(\{x_{1}\})\right)_{x_{1}^{\prime},x_{1}\in X_{1}}$$
is the matrix representing $\phi$ according to Lemma \ref{fiofhoiwefhwef32r}.  As a consequence of  Lemma \ref{fiofhoiwefhwef32r} we obtain:
\begin{kor}\label{fuewihfiuwehfewfewfwef}A matrix \eqref{f34flk3n4fok34f34f}  represents a morphism \eqref{f3oifjiocvcwecw}
 iff the following conditions are satisfied:
\begin{enumerate}
\item There exists an entourage $U_{1}$ in $\cC( X_{1})$ such that for every $x_{0}$ in $X_{0}$ and $x_{1}$ in $X_{1}$ the condition
$\phi^{f}_{x_{1},x_{0}}\not=0$ implies that $(x_{1},f(x_{0}) )\in U_{1}$.
\item For every $g$ in $G$ we have the equality
$$\rho_{1} (g)_{x_{1}}\circ \phi^{f}_{x_{1} ,x_{0}}=
	(g\phi^{f})_{g^{-1}x_{1},g^{-1}x_{0}}\circ \rho(g)_{x_{0}}
\ .$$
\end{enumerate}
\end{kor}

\subsection{The symmetric monoidal  refinement of $\bV_{{\mathbf{A}}}^{G}$}\label{vevgebvveerverv}

\newcommand{\mathcal{V}_{\bA}^{G}}{\mathcal{V}_{{\mathbf{A}}}^{G}}

Let ${\mathbf{A}}$ be a small  additive category with a strict $G$-action.
Then we let $$\pi:\mathcal{V}_{\bA}^{G}\to G\mathbf{BornCoarse}$$ denote the {Grothendieck} construction associated to the functor $\bV_{{\mathbf{A}}}^{G}$ {(see \eqref{ekrjekrferferferf})}  viewed as a functor from $G\mathbf{BornCoarse}$ to ${\mathbf{Cat}}_{1}$. The goal of this section is the construction of a symmetric monoidal structure (see Definition \ref{gui23r32reger}) on $\mathcal{V}_{\bA}^{G}$ which satisfies the assumptions of Theorem \ref{rgio34t34r34rerg}.

\begin{ass}\label{giwrogerfwewf}
We assume that ${\mathbf{A}}$ has a symmetric monoidal structure and that the strict  action of $G$ on ${\mathbf{A}}$ has a refinement to an action by symmetric monoidal functors.
\end{ass}

In order to introduce the notation for later arguments, we spell out the Assumption \ref{giwrogerfwewf} explicitly.
According to Definition \ref{gui23r32reger}   the category  ${\mathbf{A}}$ comes with the following data:
\begin{enumerate}
\item a bifunctor $-\otimes_{{\mathbf{A}}}-$,
\item a tensor unit $1_{{\mathbf{A}}}$,
\item an associativity constraint $\alpha^{{\mathbf{A}}}$,
\item a unit constraint $\eta^{{\mathbf{A}}}$,
\item a symmetry constraint $\sigma^{{\mathbf{A}}}$.
\end{enumerate}
This data satisfy the {relations}   specified in Definition \ref{gui23r32reger}.

The strict action of $G$ on ${\mathbf{A}}$ by symmetric monoidal functors   is implemented by the following data. For every $g$ in $G$ we have:
\begin{enumerate}
\item  an additive functor
$g:{\mathbf{A}}\to {\mathbf{A}}$,
\item\label{fweiufeiwofewfewf} an isomorphism $\epsilon^{g}:1_{{\mathbf{A}}}\to g1_{{\mathbf{A}}}$,
\item\label{vuhiwefwefwefewfwef2r} a natural isomorphism $\mu^{g}:(g-\otimes_{{\mathbf{A}}}g-)\to g(-\otimes_{{\mathbf{A}}}-)$,
\end{enumerate}
satisfying the {relations}  specified in Definition \ref{34f8924fregerg}.
We require that for all $g$ and $h $ in $G$   the following relation between the composition of symmetric monoidal functors and multiplication in $G$ holds true:
\begin{equation}\label{rgkjbkrjg34f34f34f}
(g,\epsilon^{g},\mu^{g})\circ (h,\epsilon^{h},\mu^{h})=
(gh,\epsilon^{gh},\mu^{gh})\ .
\end{equation}  The equality (as opposed to the additional data of a natural transformation)
expresses the fact that the action of $G$ on ${\mathbf{A}}$ is strict.

We now describe the category $\mathcal{V}_{\bA}^{G}$
explicitly.
\begin{enumerate}
\item The objects of $\mathcal{V}_{\bA}^{G}$ are pairs $(X,(M,\rho))$ of {objects} $X$ in $G\mathbf{BornCoarse}$ and $(M,\rho)$ in $\bV_{{\mathbf{A}}}^{G}(X)$.
\item\label{gio43f34ferfef} A morphism $(f,\phi):(X,(M,\rho))\to (X^{\prime},(M^{\prime},\rho^{\prime}))$ consists of a morphism
$f:X\to X^{\prime}$ in $G\mathbf{BornCoarse}$ and a morphism $\phi:f_{*}(M,\rho)\to (M^{\prime},\rho^{\prime})$ in $\bV_{{\mathbf{A}}}^{G}(X^{\prime})$.
\item The composition of morphisms is given by
$$(f^{\prime},\phi^{\prime})\circ (f,\phi):=(f^{\prime}\circ f,\phi^{\prime}\circ f^{\prime}_{*}(\phi))\ .$$
\end{enumerate}

The functor $$\pi:\mathcal{V}_{\bA}^{G}\to G\mathbf{BornCoarse}\ , \quad  (X,(M,\rho))\mapsto X$$ is the obvious functor
  which forgets the second component.

We now start with the description of the symmetric monoidal structure on $\mathcal{V}_{\bA}^{G}$.

  Let $*$ denote the one-point space.
Then we can  consider the  equivariant $*$-controlled ${\mathbf{A}}$-object
$1_{*}=(M^{unit},\rho^{unit})$ in $\bV_{A}^{G}(*)$
defined as follows:
\begin{enumerate}
\item The functor $M^{unit}:\cB(*)\to {\mathbf{A}}$ is uniquely determined by
$M^{unit}(\{*\}):=1_{{\mathbf{A}}}$.
\item $\rho^{unit}(g):=\epsilon^{g}$ for all $g$ in $G$ (see \ref{fweiufeiwofewfewf}.).
\end{enumerate}

\begin{ddd} \label{4gtiuo34ergg}The tensor unit of $\mathcal{V}_{\bA}^{G}$ is defined to be the object $1_{\mathcal{V}_{\bA}^{G}}:=(*,1_{*})$.
\end{ddd}

We now construct   the bifunctor \begin{equation}\label{23fb2jkbnfij23de23d23d}
-\otimes_{\mathcal{V}_{\bA}^{G}}- :\mathcal{V}_{\bA}^{G}\times \mathcal{V}_{\bA}^{G}\to \mathcal{V}_{\bA}^{G}\ .
\end{equation}
 We start with its definition on objects. We consider two objects
$(X,(M,\rho))$ and $(X^{\prime},(M^{\prime},\rho^{\prime}))$ in $\mathcal{V}_{\bA}^{G}$.
Then we define the functo
$$M\boxtimes  M^{\prime}:\cB(X\otimes X^{\prime})\to {\mathbf{A}}$$
as follows:
\begin{enumerate}
\item For every $B$ in $\cB(X\otimes X^{\prime})$ we set (see  Convention \ref{choosesum})
$$(M\boxtimes  M^{\prime})(B):=\bigoplus_{(x,x^{\prime})\in B} M(\{x\})\otimes_{{\mathbf{A}}}M^{\prime}(\{x^{\prime}\})\ .$$
Note that the sum has finitely many non-zero summands because of Definition \ref{def:Xcontrolledobject} (\ref{def:Xcontrolledobject:it4}).
\item If $B^{\prime}$ is in $\cB(X\otimes X^{\prime})$ such that $B^{\prime}\subseteq B$, then the morphism
$$(M\boxtimes  M^{\prime})(B^{\prime}\subseteq B)\colon(M\boxtimes  M^{\prime})(B^{\prime})\to
(M\boxtimes  M^{\prime})(B)$$
is given by the canonical map
$$\bigoplus_{(x,x^{\prime})\in B^{\prime}}
M(\{x\})\otimes_{{\mathbf{A}}}M^{\prime}(\{x^{\prime}\})
\to \bigoplus_{(x,x^{\prime})\in B }
M(\{x\})\otimes_{{\mathbf{A}}}M^{\prime}(\{x^{\prime}\})$$
as described in Convention \ref{choosesum}.
\end{enumerate}
 By using our Convention \ref{choosesum} and the universal property of the direct sum, one easily checks that this describes a functor satisfying the first three conditions of Definition \ref{def:Xcontrolledobject}.

 We now define the family
 $\rho\boxtimes \rho^{\prime}$
 as follows:
 $$\hspace{-1cm}(\rho\boxtimes \rho^{\prime})(g)_{B}:= \oplus_{(x,x^{\prime})\in B} \rho(g)_{x}\otimes \rho^{\prime}(g)_{x^{\prime}}:
  \bigoplus_{(x,x^{\prime})\in B }
M(\{x\})\otimes_{{\mathbf{A}}}M^{\prime}(\{x^{\prime}\})\to \bigoplus_{(x,x^{\prime})\in B }
gM(\{g^{-1}x\})\otimes_{{\mathbf{A}}}gM^{\prime}(\{g^{-1}x^{\prime}\})$$
using the notation \eqref{v4rr4vhu34fr3f}.
One checks using \eqref{rgkjbkrjg34f34f34f}  that
$(M\boxtimes M^{\prime},\rho\boxtimes \rho^{\prime})$  satisfies the remaining condition of  Definition \ref{def:Xcontrolledobject}
 and therefore belongs to $\bV_{{\mathbf{A}}}^{G}(X\otimes X^{\prime})$.

 \begin{ddd}
We define the bifunctor \eqref{23fb2jkbnfij23de23d23d} on objects by
$$(X,(M,\rho))\otimes_{\mathcal{V}_{\bA}^{G}}(X^{\prime},(M^{\prime},\rho^{\prime})):=(X\otimes X^{\prime},(M\boxtimes M^{\prime},\rho\boxtimes \rho^{\prime}))\ .$$
 \end{ddd}

Let $(f,\phi):(X_{0},(M_{0},\rho_{0}))\to (X_{1},(M_{1},\rho_{1}))$ be a morphism in $\mathcal{V}_{\bA}^{G}$ (see \ref{gio43f34ferfef}).
Then we define the morphism
$$\hspace{-1cm}(g,\psi):=(f,\phi)\otimes_{\mathcal{V}_{\bA}^{G}} (X^{\prime},(M^{\prime},\rho^{\prime})):  (X_{0}\otimes X^{\prime} ,(M_{0}\boxtimes M^{\prime}, \rho_{0}\boxtimes \rho^{\prime}))\to  (X_{1}\otimes X^{\prime} ,(M_{1}\boxtimes M^{\prime}, \rho_{1}\boxtimes \rho^{\prime}))$$ as follows.
\begin{enumerate}
\item We set $g:=f\otimes \id_{X^{\prime}}:X_{0}\otimes X^{\prime}\to X_{1}\otimes X^{\prime}$ using the tensor product in $G\mathbf{BornCoarse}$.
\item In order to describe the morphism
$$\psi:(f\otimes\id_{X^{\prime}})_{*} (M_{0}\boxtimes M^{\prime}, \rho_{0}\boxtimes \rho^{\prime})\to (M_{1}\boxtimes M^{\prime}, \rho_{1}\boxtimes \rho^{\prime})$$ we use Corollary \ref{fuewihfiuwehfewfewfwef}. We must describe the matrix
$$(\psi^{f\otimes \id_{X^{\prime}}}_{(x_{1},y^{\prime}),(x_{0},x^{\prime})})_{(x_{0},x^{\prime})\in X_{0}\times X^{\prime}, (x_{1},y^{\prime})\in X_{1}\times X^{\prime}}\ .$$
Now note that by definition
$$(M_{0}\boxtimes M^{\prime})(\{x_{0},x^{\prime}\})\cong M_{{0}}(\{x_{0}\})\otimes_{{\mathbf{A}}} M^{\prime}(\{x^{\prime}\})$$ so that we can set
$$\psi^{f\otimes  \id_{X^{\prime}}}_{(x_{1},y^{\prime}),(x_{0},x^{\prime})}:=\phi^{f}_{x_{1},x_{0}}\otimes_{{\mathbf{A}}} (\id_{(M^{\prime},\rho^{\prime})})_{y^{\prime},x^{\prime}} :M_{0}(\{x_{0}\})\otimes_{{\mathbf{A}}} M^{\prime}(\{x^{\prime}\})\to M_{1}(\{x_{1}\})\otimes_{{\mathbf{A}}} M^{\prime}(\{y^{\prime}\})\ .$$
 One easily checks that
this matrix satisfies the conditions listed in  Corollary \ref{fuewihfiuwehfewfewfwef}  and therefore represents the desired morphism.
\end{enumerate}
In a similar manner we define
$(X,(M,\rho))\otimes (f^{\prime},\phi^{\prime})$
for a morphism
 $(f^{\prime},\phi^{\prime}):(X^{\prime}_{0},(M^{\prime}_{0},\rho^{\prime}_{0}))\to (X^{\prime}_{1},(M^{\prime}_{1},\rho_{1}))$.

\begin{ddd}
We define the bifunctor \eqref{23fb2jkbnfij23de23d23d}
on morphisms by the preceding description.
\end{ddd}

It is straightforward to check that \eqref{23fb2jkbnfij23de23d23d} is  a bifunctor, i.e., that its description on morphisms is compatible with composition.

Next we define the associativity constraint
$\alpha^{\mathcal{V}_{\bA}^{G}}$. We consider three objects
$(X,(M,\rho))$, $(X^{\prime},(M^{\prime},\rho^{\prime}))$, and
$(X^{\prime\prime},(M^{\prime\prime},\rho^{\prime\prime}))$. Then
$$ (f,\phi):=
\alpha_{(X,(M,\rho)),(X^{\prime},(M^{\prime},\rho^{\prime})),(X^{\prime\prime},(M^{\prime\prime},\rho^{\prime\prime}))}$$ must be a  morphism
$$
((X\otimes X^{\prime})\otimes X^{\prime\prime},((M\boxtimes M^{\prime})\boxtimes M^{\prime\prime},(\rho\boxtimes \rho^{\prime})\boxtimes \rho''))\to (X\otimes (X^{\prime}\otimes X^{\prime\prime}),(M\boxtimes (M^{\prime}\boxtimes M^{\prime\prime}),\rho\boxtimes (\rho^{\prime}\boxtimes \rho'')))$$
We set
$$f:=\alpha_{X,X^{\prime},X^{\prime\prime}}$$
using the associativity constraint of $G\mathbf{BornCoarse}$.
The second component $\phi$ is given via Corollary \ref{fuewihfiuwehfewfewfwef}  by the matrix whose only non-trivial entries are
$$\phi^{f}_{(x,(x^{\prime},x^{\prime\prime})), ((x,x^{\prime}),x^{\prime\prime})}:=\alpha_{M(\{x\}),M^{\prime}(\{x^{\prime}\}),M^{\prime\prime}(\{x''\})}$$
using the associativity constraint of ${\mathbf{A}}$.
The first condition  of Corollary \ref{fuewihfiuwehfewfewfwef} is satisfied for the diagonal entourage of $X\times (X^{\prime}\times X'')$, and for the second condition we use that $G$ acts on ${\mathbf{A}}$ by symmetric monoidal functors, in particular the first relation in Definition \ref{34f8924fregerg} for $\mu^{g}$ for all $g$ in $G$, see \ref{vuhiwefwefwefewfwef2r}.

\begin{ddd}
We define  the associativity constraint
$\alpha^{\mathcal{V}_{\bA}^{G}}$ by the description above.
\end{ddd}

It is straightforward but tedious to check that $\alpha^{\mathcal{V}_{\bA}^{G}}$ is a natural transformation.

Following Definition \ref{4gtiuo34ergg} the unit constraint $\eta^{\mathcal{V}_{\bA}^{G}}$  of $\mathcal{V}_{\bA}^{G}$ is implemented by morphisms
$$(f,\phi): (*\otimes X,  (M^{unit}\boxtimes M,\rho^{unit}\boxtimes \rho))\to (X,(M,\rho))$$ for all objects $(X,(M,\rho))$ of $\mathcal{V}_{\bA}^{G}$. We set
$$f:=\eta_{X}$$
using the unit constraint of $G\mathbf{BornCoarse}$. Note that
$$(M^{unit}\boxtimes M)(\{(*,x)\})\cong 1_{{\mathbf{A}}}\otimes_{{\mathbf{A}}} M(\{x\})\ .$$
Hence, using  Corollary \ref{fuewihfiuwehfewfewfwef}, we can define  morphism $\phi$ such that the non-trivial entries of its matrix are
$$\phi^{f}_{x,(*,x)}:=\eta_{M(\{x\})}$$
using the unit constraint of ${\mathbf{A}}$. It is easy to check that this matrix satisfies the first condition of Corollary \ref{fuewihfiuwehfewfewfwef} for the diagonal of $X$
and the second condition since the morphisms $\epsilon^{g}$ in \ref{fweiufeiwofewfewf} satisfy the relation {of Definition }\ref{34f8924fregerg}.\ref{fiowfewef23rr3} for all $g$ in $G$.

\begin{ddd}
We define the unit constraint $\eta^{\mathcal{V}_{\bA}^{G}}$ {by} the description above.
\end{ddd}

It is straightforward to check that $\eta^{\mathcal{V}_{\bA}^{G}}$
is a natural transformation.

Finally we define the symmetry constraint $\sigma^{\mathcal{V}_{\bA}^{G}}$.
We consider two objects $(X,(M,\rho))$ and $(X^{\prime},(M^{\prime},\rho^{\prime}))$ of $\mathcal{V}_{\bA}^{G}$.
Then we must define a morphism
$$(f,\phi):(X\otimes X^{\prime},(M\boxtimes M^{\prime},\rho\boxtimes \rho^{\prime}))\to
(X^{\prime}\otimes X ,(M^{\prime}\boxtimes M ,\rho^{\prime}\boxtimes \rho ))\ .$$
We set
$$f:=\sigma_{X,X^{\prime}}$$ using the symmetry contraint for $G\mathbf{BornCoarse}$. The morphism
$\phi$ is the given, using Corollary \ref{fuewihfiuwehfewfewfwef}, by the matrix whose only non-trivial entries are
$$\phi^{f}_{(x^{\prime},x),(x,x^{\prime})}:=\sigma_{M(\{x\}),M^{\prime}(\{x^{\prime}\})}$$
using the symmetry constraint of ${\mathbf{A}}$.
One easily {checks} that the first condition of
 Corollary \ref{fuewihfiuwehfewfewfwef} is satisfied for the diagonal entourage of $X^{\prime}\times X$. In order to verify the second condition we use that
 the transformations $\mu^{g}$ in \ref{vuhiwefwefwefewfwef2r}
 satisfy {Definition} \ref{34f8924fregerg}.\ref{evfiwehfioewfewfewfewff} for every $g$ in $G$.

\begin{ddd}
We define the symmetry constraint  $\sigma^{\mathcal{V}_{\bA}^{G}}$  of $\mathcal{V}_{\bA}^{G}$ by the description above.
\end{ddd}

It is straightforward to check that $\sigma^{\mathcal{V}_{\bA}^{G}}$
is a natural transformation.

\begin{prop}\label{efiweufowefewfw}
The functor $-\otimes_{\mathcal{V}_{\bA}^{G}}-$ and the object $1_{\mathcal{V}_{\bA}^{G}}$ together with the natural isomorphisms
$\alpha^{\mathcal{V}_{\bA}^{G}}$, $\eta^{\mathcal{V}_{\bA}^{G}}$ and $\sigma^{\mathcal{V}_{\bA}^{G}}$
define a symmetric monoidal structure on $\mathcal{V}_{\bA}^{G}$.

The functor
$\pi:\mathcal{V}_{\bA}^{G}\to G\mathbf{BornCoarse}$ preserves the tensor product and   the tensor unit as well as the associator, unit, and symmetry transformations.
 \end{prop}
 \begin{proof}
 One verifies the relations listed in Definition \ref{gui23r32reger} in a straightforward manner by inserting the definitions and using that the corresponding relations are satisfied for the symmetric monoidal structures on ${\mathbf{A}}$ and $G\mathbf{BornCoarse}$.
 \end{proof}

Let $X$ and $X^{\prime}$ be $G$-bornological coarse spaces.
\begin{prop}\label{efiweufowefewfw1}
The functor
$$\boxtimes_{X,X^{\prime}}:\bV^{G}_{{\mathbf{A}}}(X)\times \bV^{G}_{{\mathbf{A}}}(X^{\prime})\to \bV_{{\mathbf{A}}}^{G}(X{\otimes} X^{\prime})$$ obtained in \eqref{f3foi34jfio34jf34f34f3f} is additive in both variables.
\end{prop}
\begin{proof}
Let $(M_{i},\rho_{i})$ be in $\bV_{{\mathbf{A}}}^{G}(X)$ for $i=0,1$  and $(M^{\prime},\rho)$ be in $\bV_{{\mathbf{A}}}^{G}(X^{\prime})$.  In view of the symmetry it suffices to show that the canonical morphism
$$(M_{0}\boxtimes_{X,X^{\prime}}M^{\prime})\oplus (M_{1}\boxtimes_{X,X^{\prime}}M^{\prime})\to (M_{0}\oplus M_{1})\boxtimes_{X,X^{\prime}} M^{\prime}$$
is an isomorphism. In view of Conditions
 \ref{def:Xcontrolledobject}.\ref{def:Xcontrolledobject:it1} and \ref{def:Xcontrolledobject}.\ref{def:Xcontrolledobject:it3} it suffices to show that
$$\left[(M_{0}\boxtimes_{X,X^{\prime}}M^{\prime} )\oplus (M_{1}\boxtimes_{X,X^{\prime}}M^{\prime})\right](\{(x,x^{\prime})\})\to \left[(M_{0}\oplus M_{1})\boxtimes_{X,X^{\prime}} M^{\prime}\right](\{(x,x^{\prime})\})$$
is an isomorphism for every point $(x,x^{\prime})$ in $X\times X^{\prime}$.
By inserting the definitions we see that  this morphism is the same as
$$(M_{0}(\{x\})\otimes_{{\mathbf{A}}} M^{\prime}(\{x^{\prime}\}))\oplus (M_{1}(\{x\})\otimes_{{\mathbf{A}}} M^{\prime}(\{x^{\prime}\}))\to (M_{0}(\{x\})\oplus M_{1}(\{x\}))\otimes_{{\mathbf{A}}} M^{\prime}(\{x^{\prime}\})\ .$$
 But this last morphism is an isomorphism since the tensor product in ${\mathbf{A}}$ is additive in the first argument.
\end{proof}

Let $f\colon X\to X^{\prime}$ and $f^{\prime}\colon Y\to Y^{\prime}$ be two morphisms of $G$-bornolgical coarse spaces. Let $(M,\rho)$  be in $\bV_{{\mathbf{A}}}^{G}(X)$
and $(N,\eta)$ be in $\bV_{A}^{G}(Y)$.

\begin{lem}\label{lemmaiso} The morphism   	 	$$(f\otimes f^{\prime})_{*}((M,\rho)\boxtimes_{X ,Y} (N,\eta)) \to  f_{*}(M,\rho)\boxtimes_{X^{\prime},Y^{\prime}} f^{\prime}_{*}(N,\eta)$$  in $\bV_{{\mathbf{A}}}^{G}(X^{\prime}\otimes Y^{\prime})$ (see {\eqref{vwrkjbwwejkvjbnkwevewvw}})
 is an isomorphism.
\end{lem}
\begin{proof}
In view of Conditions
 \ref{def:Xcontrolledobject}.\ref{def:Xcontrolledobject:it1} and \ref{def:Xcontrolledobject}.\ref{def:Xcontrolledobject:it3} it suffices to show that
 $$[(f\otimes f^{\prime})_{*}((M,\rho)\boxtimes_{X,Y} (N,\eta))](\{(x^{\prime},y^{\prime})\}) \to  [f_{*}(M,\rho)\boxtimes_{X^{\prime},Y^{\prime}} f^{\prime}_{*}(N,\eta)](\{(x^{\prime},y^{\prime})\})$$
 is an isomorphism for every point $(x^{\prime},y^{\prime})$ in $X^{\prime}\times Y^{\prime}$. Inserting the definitions this morphism is given by
 \begin{equation}\label{fehi23ufhwefwfefe}\bigoplus_{(x,y)\in (f\times f^{\prime})^{-1}(\{x^{\prime},y^{\prime}\})} M(\{x\})\otimes_{{\mathbf{A}}} N(\{y\})\to  \left(\bigoplus_{x\in f^{-1}(\{x^{\prime}\})} M(\{x\})\right) \otimes_{{\mathbf{A}}} \left(\bigoplus_{y\in (f^{\prime})^{-1}(\{y^{\prime}\})} N(\{y\})\right)\end{equation}
 which for every $(x,y)$ in  $(f\times f^{\prime})^{-1}(\{x^{\prime},y^{\prime}\})$
 is the morphism
  $$  M(\{x\})\otimes_{{\mathbf{A}}} N(\{y\})\to  \left(\bigoplus_{x\in f^{-1}(\{x^{\prime}\})} M(\{x\})\right) \otimes_{{\mathbf{A}}} \left(\bigoplus_{y\in (f^{\prime})^{-1}(\{y^{\prime}\})} N(\{y\})\right) $$
  induced by the inclusions of the respective summands of the tensor factors.
  Since the tensor product in ${\mathbf{A}}$ preserves sums in both arguments we conclude that
 \eqref{fehi23ufhwefwfefe} is an isomorphism.
\end{proof}

In view of Theorem \ref{rgio34t34r34rerg} the  Propositions \ref{efiweufowefewfw} and \ref{efiweufowefewfw1} and Lemma \ref{lemmaiso} now imply:

\begin{theorem}\label{mainthm}
If ${\mathbf{A}}$ is a symmetric monoidal  additive category with a strict action of $G$ by symmetric monoidal functors, then the functor $loc\circ \bV_{{\mathbf{A}}}^{G}:G\mathbf{BornCoarse}\to  {\mathtt{Add}}_{\infty}$ admits a refinement to a {lax} symmetric monoidal functor
$$\bV_{{\mathbf{A}}}^{G,\otimes}:\Nerve(G\mathbf{BornCoarse}^{\otimes})\to {\mathtt{Add}}_{\infty}^{\otimes}\ .$$
\end{theorem}

\subsection{The symmetric monoidal $K$-theory functor for additive categories}\label{foiwejfowfwefefwef}

\newcommand{\mathbf{dgCat}}{\mathbf{dgCat}}

In \cite{buci} a universal $K$-theory functor $$\mathrm{UK}:\Nerve({\mathtt{Add}}_{1})\to \cM_{loc}$$
was considered, where $\cM_{loc}$ is the category of non-commutative motives of Blumberg-Gepner-Tabuada \cite{MR3070515}. This functor was defined {as the upper horizontal composition  in the diagram}
$$\xymatrix{\Nerve({\mathtt{Add}}_{1})\ar[d]^{loc}
\ar[r]^{\Ch^{b}[-]_{\infty}}&{\mathbf{Cat}}^{ex}_{\infty}\ar[r]^{{\mathcal{U}}_{loc}}& \cM_{loc}\\{\mathtt{Add}}_{\infty}\ar@{..>}[urr]_{\mathrm{UK}_{\infty}}&&
}\ ,$$
where ${\mathcal{U}}_{loc}$ is the universal localizing invariant, and $\Ch^{b}[-]_{\infty}$ sends an additive category ${\mathbf{A}}$ to the stable $\infty$-category of bounded chain complexes over ${\mathbf{A}}$ with homotopy equivalences   inverted.
Since the functor $\mathrm{UK}$ preserves equivalences of additive categories we have the indicated factorization ${\mathrm{UK}_{\infty}}$.

  \begin{theorem}\label{rgreioghjo34tergergeg21}
The functor $\mathrm{UK}_{\infty}$ admits a symmetric monoidal refinement
$$\mathrm{UK}_{\infty}^{\otimes}:{\mathtt{Add}}_{\infty}^{\otimes}\to \cM_{loc}^{\otimes}\ .$$
\end{theorem}
\begin{proof}
 The proof of this theorem will be finished at the end of the present section.
 As a first step we observe that it suffices to construct a symmetric monoidal refinement  $\mathrm{UK}^{\otimes}$ of $\mathrm{UK}$.
 Then we obtain the symmetric monoidal refinement  $\mathrm{UK}^{\otimes}_{\infty} $ of $\mathrm{UK}_{\infty}$ from the universal property of the symmetric monoidal localization $loc^{\otimes}:\Nerve({\mathtt{Add}}_{1}^{\otimes})\to {\mathtt{Add}}_{\infty}^{\otimes}$ using \cite{hinich}.

By   \cite{bgt-2} the universal localizing invariant ${\mathcal{U}}_{loc}$  refines to a symmetric monoidal functor
$${\mathcal{U}}^{\otimes}_{loc}:{\mathbf{Cat}}^{ex,\otimes}_{\infty}\to \cM^{\otimes}_{loc}\ .$$  It therefore remains to produce a symmetric monoidal functor $$St^{\otimes}: {\mathtt{Add}}^{\otimes}_{\infty} \to {\mathbf{Cat}}^{ex,\otimes}_{\infty}$$ refining $\Ch^{b}[-]_{\infty}$. We use the symbol $St$ in order to indicate that this functor is related  with stabilization.

We are going to use the following notation.
 The category $\mathbf{dgCat}_{1}$ is the $1$-category of {small} dg-categories. The  set  $W_{Morita}$ is  the set of Morita equivalences, i.e., functors between $dg$-categories $\cC\to {\mathcal{D}}$ which induce an equivalence of derived categories \cite[Sec. 4.6]{MR2275593},  \cite[Def. 2.29]{cohn}.

 The category $\mathbf{dgCat}_{1}$  contains the full subcategory
$\mathbf{dgCat}_{1,flat}$  of locally flat dg-categories, i.e., dg-categories $\bC$ with the property that for every two objects $C,C^{\prime}$ in $\bC$ the complex $\Hom_{\bC}(C,C^{\prime})$ consists of flat $\Z$-modules.  It furthermore contains the full subcategory of pre-triangulated $dg$-categories \cite[Sec. 4.5]{MR2275593}, \cite{BKdg}.

Furthermore, ${\mathbf{Cat}}^{ex}_{\infty,H\Z}$ is the category of $H\Z$-linear stable {idempotent complete} $\infty$-categories and $H\Z$-linear exact functors, and $\cF$ forgets the $H\Z$-linear structure.  For the equivalence  marked by $DK$ (for Dold-Kan) we refer to \cite{cohn}.
\begin{prop}
We have the  bold part of the following commuting diagram:  \begin{equation}\label{fhwjkefkj23r23r23}
\xymatrix{&\Nerve(\mathbf{dgCat}_{1}^{pre})\ar[d]^{\subseteq}\ar[r]^{\ell}&\Nerve(\mathbf{dgCat}_{1}^{pre})[W^{-1}_{Morita}]\ar[d]^{\simeq}_{\iota} \ar[drr]^{\Nerve_{\infty}^{dg}(-)}&&\\\Nerve({\mathtt{Add}}_{1})\ar@/^4cm/@{..>}[rrrr]^{St}\ar@{..>}[dr]^{\mathbf{Q}}\ar[r]&\Nerve(\mathbf{dgCat}_{1})\ar[ur]^{\ell\circ \Ch^{b}}\ar[r]^{\ell}&\Nerve(\mathbf{dgCat}_{1})[W^{-1}_{Morita}]\ar[r]_{\simeq}^{DK}& {\mathbf{Cat}}_{\infty,H\Z}^{ex}\ar[r]_{\cF}&{\mathbf{Cat}}^{ex}_{\infty}\\ &\Nerve(\mathbf{dgCat}_{1,flat})\ar[u]\ar[r]&\Nerve(\mathbf{dgCat}_{1,flat})[W^{-1}_{Morita}]\ar[ur]\ar[u]_{!!}^{\simeq}& &}\ .\end{equation}
\end{prop}
\begin{proof}
\begin{enumerate}
\item For every $dg$-category the canonical inclusion  $\cC\to \Ch^{b}(\cC)$  represents the pretriangulated hull \cite[Sec. 4.5]{MR2275593}, \cite{BKdg}. In particular, the functor $\Ch^{b}$  has values in pretriangulated $dg$-categories.
\item The two triangles in the corresponding square commute since
  for every $dg$-category $\cC$ the canonical inclusion induces a Morita equivalence $\cC\to \Ch^{b}(\cC)$. To this end we
   use that the inclusion of $\cC$ into its triangulated hull is a Morita {equivalence} {\cite[Sec. 4.6]{MR2275593}}.

\item The $dg$-nerve $\Nerve^{dg} :\Nerve(\mathbf{dgCat}_{1}^{pre})\to {\mathbf{Cat}}^{ex}_{\infty}$ preserves Morita equivalences and therefore descends to $\Nerve^{dg}_{\infty}$ as indicated.
\item We have an equivalence $\cF\circ DK \circ  \iota\simeq \Nerve_{\infty}^{dg}$, \cite[Prop. 3.3.2]{MR3607208}, see also \cite{Tabuada_2010}, \cite{toen}.
In order to provide more details we consider   the functor $Z^{0}:\mathbf{dgCat}_{1}^{pre}\to {\mathbf{Cat}}_{\infty}$   which associates to a $dg$-category  its underlying category (with $\Hom_{Z^{0}(\cC)}(A,B)=Z^{0}(\Hom_{\cC}(A,B))$) considered as an $\infty$-category. We  furthermore let $W_{\cC}$ be the morphisms in $Z^{0}(\cC)$ which become isomorphisms in the homotopy category $H^{0}(\cC)$ (with $\Hom_{H^{0}(\cC)}(A,B)=H^{0}(\Hom_{\cC}(A,B))$).
 Then both functors
 $$Z^{0}(\cC)\to N^{dg}(\cC) \ ,\quad
 Z^{0}(\cC)\to \cF(DK(\iota(\cC)))$$ present
 the localization $ Z^{0}(\cC)\to Z^{0}(\cC)[W_{\cC}^{-1}]$.

\end{enumerate}

\end{proof}

  The horizontal composition given by the  middle row  in \eqref{fhwjkefkj23r23r23}  defines a  functor $St$.

\begin{lem}
The functor $St$ is equivalent to the functor
$\Ch^{b}(-)_{\infty}$ constructed in \cite[Prop. 2.11]{buci}.
\end{lem}
\begin{proof}
By \cite[Rem. 2.9]{buci} we have the first equivalence of functors in the chain
$$\Ch^{b}(-)_{\infty}\simeq \Nerve^{dg}\circ \Ch^{b}(-)\simeq \Nerve^{dg}_{\infty}\circ \ell\circ \Ch^{b}$$ from ${\mathtt{Add}}_{1}$ to ${\mathbf{Cat}}^{ex}_{\infty}$. This implies the Lemma in view of the commutativity of
\eqref{fhwjkefkj23r23r23}.
\end{proof}

\begin{prop} The functor $St$ has a symmetric monoidal refinement $St^{\otimes}$.
\end{prop}
\begin{proof}
All   $1$-categories in the lower {two} lines of the diagram \eqref{fhwjkefkj23r23r23} have symmetric monoidal structures and the functors
connecting them have canonical symmetric monoidal refinements. The same is true for the $\infty$-categories and the remaining functors except  for
$\Nerve(\mathbf{dgCat}_{1})[W^{-1}_{Morita}]$ and the corresponding functors.
The problem is that the tensor product of dg-categories is not compatible with Morita equivalences and therefore does not descend to the localization directly. For this reason one considers the
subcategory of locally flat dg-categories and uses the equivalence $!!$ in order to transfer the symmetric monoidal structures. So in order to construct the symmetric monoidal refinement of
$St$ we must bypass this node of the diagram. To this end we use a symmetric monoidal flat resolution functor $\mathbf{Q}$ as indicated. The left triangle in  \eqref{fhwjkefkj23r23r23}  is filled by a natural transformation (not an isomorphism), but
 the square
$$\xymatrix{N({\mathtt{Add}}_{1})\ar[rr]\ar[dr]^{\mathbf{Q}}&&\Nerve(\mathbf{dgCat}_{1})[W^{-1}_{Morita}]\\&\Nerve(\mathbf{dgCat}_{1,flat})\ar[r]&\Nerve(\mathbf{dgCat}_{1,flat})[W^{-1}_{Morita}]\ar[u]^{\simeq}_{!!}
}$$ {does
commute}. We then get the following commuting diagram of symmetric monoidal functors
 $$\xymatrix{  \Nerve({\mathtt{Add}}_{1}^{\otimes})\ar@/^1.5cm/[rrrr]^{St^{\otimes}}\ar[dr]^{\mathbf{Q}^{\otimes}}&&& {\mathbf{Cat}}_{\infty,H\Z}^{ex,\otimes}\ar[r]^{\cF}&{\mathbf{Cat}}^{ex,\otimes}_{\infty}\\&\Nerve(\mathbf{dgCat}_{1,flat}^{\otimes})\ar[r]&\Nerve(\mathbf{dgCat}_{1,flat})^{\otimes}[W^{-1}_{Morita}]\ar[ur]& &}$$
defining the symmetric monoidal refinement $St^{\otimes}$ of $St$.

It remains to argue that a  symmetric monoidal  flat resolution functor $\mathbf{Q}$ exists.
We start with the following well-known fact.
\begin{lem}\label{giojwggrgreg}
\begin{enumerate}
\item There exists a functor $Q$ fitting into the commuting diagram
$$\xymatrix{&\Ch_{flat}\ar[d]\\\Ab\ar[ur]^{Q}\ar[r]^{(-)[0]}&\Ch}$$
such that the filler is a quasi-isomorphism.
\item The functor $Q$ has a lax symmetric monoidal structure.
\end{enumerate}
\end{lem}
\begin{proof} The natural idea works. The functor $Q$ sends   $A$ in $\Ab$ to $$Q(A):=(F_{1}(A)\stackrel{d_{A}}{\to} F_{0}(A))$$ in $\Ch$, where $F_{0}(A):=\Z[A]$ is the free abelian group generated by the underlying set of $A$, $F_{1}(A)$ is the kernel of the canonical homomorphism $F_{0}(A)\to A$, and $d_{A}$ is the inclusion.
\end{proof}

We define the flat resolution functor for additive categories by
$$\xymatrix{&&\mathbf{dgCat}_{1,flat}\ar[d]\\{\mathtt{Add}}_{1}\ar[urr]^{\mathbf{Q}}\ar[r]&{\mathbf{Cat}}_{\Ab}\ar[r]\ar@{.>}[ur]&\mathbf{dgCat}_{1}}$$
where the dotted arrow is the natural functor induced from the lax symmetric monoidal functor $Q$
which provides a functor from $\Ab$-enriched categories to $\Ch$-enriched categories with flat $\Hom$-complexes. Furthermore, the symmetric monoidal structure on $Q$ induces naturally a symmetric monoidal structure on $\mathbf{Q}$.
\end{proof}

This finishes the proof of Theorem \ref{rgreioghjo34tergergeg21}.
\end{proof}

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


text:
Ben Gibson Appointed to the First District Court of Appeal Judicial Nominating Commission
On July 17, Governor Rick Scott announced Tallahassee partner, Benjamin J. Gibson was appointed to the First District Court of Appeal Judicial Nominating Commission (JNC). Ben is appointed for a four-year term ending on July 1, 2022.
The JNC is charged with identifying and recommending names of qualified individuals to the Governor for appointment to the bench. The JNC investigates each applicant to confirm eligibility and interviews applicants.
The First District Court of Appeal is one of the largest appellate courts in the country both in terms of judges and number of cases filed annually.
To view the full list of appointments and reappointments, click here.
About Benjamin J. Gibson
Benjamin Gibson is a partner in the Tallahassee office of Shutts & Bowen LLP, where he is a member of the Business Litigation practice group. His statewide practice focuses on government, political, and corporate clients representing them in government affairs, appellate, litigation, and administrative matters. Prior to joining Shutts, Ben served as the Deputy General Counsel and an Assistant General Counsel to Florida Governor Rick Scott. He also served as a staff attorney with the Florida House of Representatives.
View Related Professionals

redpajama_set_name: RedPajamaCommonCrawl


Am I reading what I think I'm reading? Another parcel tax increase ($450) to fund the schools?
I'm sick and tired of hearing that this tax increase is needed to maintain the high property values of homes in the La Cañada school district. Enough with this lie. At this rate, property taxes will surpass property value increases and will, one day, lead to a decline in values because who is going to buy a home burdened with insane taxes?
It will place an undue burden on fixed-income folks who will be forced to sell just to satisfy the insatiable appetites of the greedy unions. If anyone employed in the LCUSD feels they are underpaid, show them the door. I'm sure there is a standing line of more appreciative teachers hoping to get into one of these posh teaching positions in the La Cañada system. How many of these teachers/administrators live in La Cañada? I'll guess most don't, so they don't care if property taxes are raised.
Why are only property owners burdened with the task of funding the schools? Renters, businesses and, yes, even teachers bring their kids into the system free of this burden. If you want to raise revenue, then charge everyone equally. Either that, or allow only property owners to vote on whether or not they want their taxes raised. I live in the Sagebrush area. I was, earlier, a very strong proponent for incorporating the Sagebrush into the La Cañada school system. I am now in favor of keeping the status quo because I do not wish to become a victim of the parcel tax invasion.


text:
Morgan challenges government back benchers to vote down Finance Bill
Responding to the publication of the Finance Bill Sinn Féin Finance Spokesperson Arthur Morgan TD said Sinn Féin would oppose the Bill which unjustly targets low and middle income earners. He challenged government back benchers to have the courage to vote against what is a fundamentally flawed piece of legislation.
Deputy Morgan said, "This Finance Bill gives effect to the proposals announced in the emergency budget. The Government continues to mishandle the economic crisis. Its incompetence sees it lash out at low and middle income earners and attempt to get them to pay for a mess not of their making.
"Cuts targeted at families with children and those struggling with high mortgages have created enormous hardship. Subjecting those on low incomes to the regressive income levy is unjust and unwarranted.
"The Government has shown no interest in dealing with jobs crisis and their attempts to deal with the banking crisis have been flawed and ineffective. They appear more concerned with the plight of bankers and developers than that of ordinary people who have lost their jobs and who cannot meet their mortgage repayments. The uncertainty over what is happening in relation to mortgage interest relief has created widespread anxiety. How this has been handled is nothing short of incompetence.
"By taking more money off low and middle income workers the Government will erode consumer confidence and further undermine the economy. This bill needs to be opposed. It can be defeated. I am calling on government back benchers to have the courage to vote against what is a fundamentally flawed piece of legislation". ENDS


text:
Carlos Hathcock: The Legendary Marine Sniper
Larry Slawson
Larry Slawson received his Masters Degree at UNC Charlotte. He specializes in Russian and Ukrainian History.
Carlos Hathcock in shooting position. | Source
Carlos Hathcock: Biographical Facts
Birth Name: Carlos Norman Hathcock II
Place of Birth: Little Rock, Arkansas
Date of Death: 22 February 1999 (Fifty-Six Years of Age)
Place of Death: Virginia Beach, Virginia
Cause of Death: Complications from Multiple Sclerosis
Place of Burial: Woodlawn Memorial Gardens, Norfolk, Virginia
Spouse(s): Jo Winstead
Children: Carlos Norman Hathcock III (son)
Father: Carlos Hathcock
Mother: Agnes Hathcock
Occupation(s): Sniper in the United States Marine Corp
Military Service: 1959-1979 (First Marine Division)
Highest Rank Achieved: Gunnery Sergeant
Best Known For: Legendary Marine Sniper; 93 confirmed kills (although estimates show that he likely killed between 300 and 400 enemy troops/personnel)
Awards and Honors: Silver Star; Navy Commendation Medal, Purple Heart; Navy and Marine Corps Achievement Medal; Good Conduct Medal; National Defense Service Medal; Vietnam Service Medal; Gallantry Cross; Vietnam Campaign Medal.
Nickname(s): "White Feather"
Carlos Hathcock preparing to take a shot. | Source
Hathcock's Early Life
Carlos Norman Hathcock was born on 20 May 1942 in Little Rock, Arkansas to Carlos and Agnes Hathcock. Born into extreme poverty, Hathcock became associated with hunting (and weapons) from an early age as it was the only way to provide food for his small family. Such associations were further bolstered by his father who had served in World War Two, and who later provided young Carlos with his first weapon, a Mauser Rifle, that he had found during the war. With his rifle in hand, Carlos was known to spend hours in the woods, acting out childhood fantasies of being a soldier; hunting imaginary Japanese soldiers in the surrounding area. From his earliest days, young Carlos was determined to join the Marines, and felt that a career in the military was the highest calling any man could aspire to.
Following the separation of his parents, Hathcock and his mother went to live with his grandmother in Wynne, Arkansas. At the age of twelve, Hathcock was given a .22-caliber J.C. Higgins rifle that proved invaluable in his hunting excursions. With a keen eye for stalking prey, along with constant practice (due to the necessity of hunting for food), Carlos's rifle skills reached expert levels within only a few years.
On his seventeenth birthday (20 May 1959), Carlos finally made his dream of joining the military a reality, and enlisted in the Marine Corps at his local recruiting office in Arkansas. It was here that the illustrious career of the legendary Marine sniper first began.
Early Military Career
After joining the Marine Corps, Hathcock's shooting skills continued to grow stronger with each passing year as he learned additional (more advanced) techniques to position himself and steady his aim. Before being deployed to Vietnam, Hathcock even entered (and won) a variety of shooting championships, securing numerous trophies with relative ease. By the age of 23, Hathcock's shooting skills were put to the ultimate test when he competed for the Wimbledon Cup, America's premier marksmanship championship. Beating out his other competitors, Hathcock returned home with his prestigious prize in hand; a testament to his remarkable skills and unparalleled talents with a rifle.
Unbeknownst to Hathcock, Major Jim Land (who would later help establish the Marine Corp Scout Sniper Program) was also present during his victory at the Wimbledon Cup. Land recognized Hathcock's potential immediately, and would later play a vital role in his transition to a sniper role only a year later.
Carlos Hathcock: Quick Facts
Quick Fact #1: Carlos Norman Hathcock II was born in Little Rock, Arkansas on 20 May 1942. After his parents separated (during the first twelve years of his life), Hathcock resided with his grandmother in Wynne, Arkansas for much of his childhood. Growing up in a poor family, Hathcock took to hunting at a very early age to help feed his family. Using a .22-caliber J.C. Higgins rifle, Hathcock would comb the woods around his home in search of food. This early exposure to guns helped young Hathcock tremendously for his future career in the Marines Corps. At the age of 17 (20 May 1959), young Hathcock enlisted at the Marine recruiting office; a decision that would have an enduring impact on the rest of his life.
Quick Fact #2: Before being deployed to South Vietnam, Hathcock met the love of his life, Jo Winstead. After receiving permission to marry her, the couple married on 10 November 1962 (Marine Corps official birthday). In his spare time, Hathcock also began to participate in shooting championships, including the famous Wimbledon Cup (which he won in 1965). In 1966, however, Hathcock found himself in route to South Vietnam, as hostilities between North and South Vietnam had begun to deteriorate rapidly. Upon arrival, Hathcock was first deployed as a military policeman, but was quickly recruited by Captain Edward James Land into a sniper platoon. Land, upon discovering that Hathcock had won the Wimbledon Cup only a year prior, was greatly impressed with the young marksman's abilities, and strongly felt that his talents were much better spent with sniping rather than police duty.
Quick Fact #3: At a time when snipers were seen as outcasts in the United States military, Captain Land was determined to change the role of snipers in warfare, as he believed that every platoon should possess at least one sharpshooter. Interested in this new concept, the Marine Corp allowed Captain Land to showcase and test the abilities of sniper units; providing Land and his new sniper platoon numerous opportunities to prove themselves along the way. In only a short amount of time, Hathcock's shooting skills were immediately put to the test against both the Viet Cong and North Vietnamese Army. In total, it is estimated that Hathcock killed between 300 to 400 enemy personnel during his tours in Vietnam, although only 93 were confirmed kills (confirmed by an acting third-party officer).
Quick Fact #4: For his actions, Hathcock quickly garnered a reputation amongst the Viet Cong and PAVN troops of the North, who referred to him as "White Feather Sniper" due to the white feather he kept inside the band of his hat. At one point, the North Vietnamese placed a $30,000 bounty on Hathcock, the highest of any bounty placed on American personnel during the war. As a result, many Marines in the area donned white feathers similar to Hathcock in order to confuse enemy snipers operating in the area.
Carlos Hathcock in later life. | Source
Quick Facts Continued...
Quick Fact #5: One of Hathcock's most famous kills involved the clash with an enemy sniper known as "The Cobra." Near Hill 55 (just southwest of Da Nang), Hathcock and his spotter, John Roland Burke, stalked the Cobra for several days after it was learned that he had killed several Marines in an attempt to lure Hathcock out of hiding (sources point out that the Cobra was sent to specifically kill Hathcock). After countless hours in the jungle, Hathcock caught a quick glint (light reflection from the enemy's sniper scope) nearby. Already in the Cobra's crosshairs, Hathcock was forced to rapidly take a shot. The shot became one of the most famous sniper kills of all time, as Hathcock's bullet went straight into the Cobra's scope, killing him instantaneously.
Quick Fact #6: Perhaps one of Hathcock's most important contributions to the Vietnam War effort can be seen with his top secret mission to kill a North Vietnamese General. After being covertly inserted into the area, Hathcock crawled a distance of 1,500 yards; a feat that took four days and three nights to complete as he could only move a few inches at a time. With no sleep and being constantly bit by bugs, Hathcock slowly eased into position. Hathcock even came face-to-face with a deadly bamboo viper at one point in his trek, barely avoiding its deadly bite. After getting into a good position, however, Hathcock spotted the general as he exited his living quarters. Without hesitation, Hathcock fired a single shot into the general's chest, killing him on impact. With enemy soldiers hot on his trail, Hathcock made a rapid escape to his extraction point without ever being spotted or wounded.
Quick Fact #7: After completing his tour of duty in 1967, Hathcock returned to Vietnam in 1969 and took command of another platoon of snipers. However, shortly into his tour, Hathcock's career with the Marine Corps came to an abrupt end alongside the famed Highway 1. On board an LVT-5, Hathcock's vehicle was struck by an anti-tank mine that engulfed the vehicle in fire. Suffering severe wounds and burns, Hathcock still managed to pull seven other Marines from the burning wreckage before passing out from his injuries. Hathcock and his fellow Marines were quickly evacuated to the hospital ship, USS Repose, then to a naval hospital in Tokyo. Hathcock's burns were so severe, however, that he was later transferred to the burn center at Brooke Army Medical Center (San Antonio, Texas). Although Hathcock remained in the Marines, he would never serve in combat again due to the extensive, lifelong injuries he sustained from the explosion.
Quick Fact #8: Following his rehabilitation, Hathcock helped to establish the Marine Corps "Scout Sniper School" at Quantico, Virginia. Here, Hathcock helped to train future Marine Corp snipers for combat, while continuing to suffer extensively from his prior injuries. In 1975, Hathcock suffered another setback, however, with the diagnosis of Multiple Sclerosis. After his health continued to decline in the months and years that followed, Hathcock received a "permanent disability separation" from the Marine Corps, forcing him into an unwanted medical discharge.
Quick Fact #9: After suffering from deep depression (and having his wife nearly leave him), Hathcock was able to overcome his separation from the Marines by picking up hobbies to keep himself busy. Shark fishing, in particular, became a particular favorite of Hathcock as it offered him both a unique and difficult challenge each time he fished. In his spare time, Hathcock also aided numerous police departments and military units (most famously, SEAL Team Six) in their courses for sniping. Hathcock tragically died at the age of fifty-six after suffering from complications resulting from multiple sclerosis. He was buried at Woodlawn Memorial Gardens in Norfolk, Virginia.
Marine Corps Emblem | Source
Fun Facts About Carlos Hathcock
Fun Fact #1: When Carlos was still a teenager, he dropped out of high school to work at the Little Rock concrete company in order to better support his family.
Fun Fact #2: Hathcock knew that he wanted to be a Marine from the time he was a young boy, and often pretended to be a Marine in the woods of Arkansas.
Fun Fact #3: In another notable moment of his career as a sniper, Hathcock was able to successfully kill a notorious enemy interrogator known as the "Apache." The Apache was well-known across Vietnam due to their brutal torture methods employed against American troops. Their death served as a tremendous morale booster for American troops.
Fun Fact #4: Hathcock held the record for the longest sniper kill after taking a nearly 2,500 yard shot with an M2 .50-caliber Browning Machinegun.
Fun Fact #5: For his efforts in saving so many Marines aboard the LVT-5 that nearly claimed his own life, Hathcock was awarded the Silver Star in the 1990s, several decades after the event transpired.
Fun Fact #6: Hathcock's son, Carlos Hathcock III, also joined the Marine Corps and retired as a Gunnery Sergeant. He also served as a member of the "Board of Governors" within the Marine Corps "Distinguished Shooters Association."
Fun Fact #7: Multiple movie scenes have been inspired by the actions of Hathcock, including both "Sniper" and the movie "Saving Private Ryan."
"The most deadly thing on a battlefield is one well-aimed shot."
— Sgt. Carlos Hathcock
Sniper Quotes
Quote #1: "The most deadly thing on a battlefield is one well-aimed shot." – Sgt. Carlos Hathcock
Quote #2: "I like shooting, and I love hunting. But I never did enjoy killing anybody. It's my job. If I don't get those bastards, then they're gonna kill a lot of these kids dressed up like Marines. That's the way I look at it." – Sgt. Carlos Hathcock
Quote #3: "One man can change the world with a bullet in the right place." – Malcom McDowell
Quote #4: "People tell me I saved hundreds and hundreds of people. But I have to tell you: it's not the people you saved that you remember. It's the ones you couldn't save. Those are the ones you talk about. Those are the faces and situations that stay with you forever." – Chris Kyle
Quote #5: "Bolt actions speak louder than words." – Craig Roberts
Do you believe that Carlos Hathcock's actions helped save American lives in Vietnam?
Carlos Hathcock Books
Chandler, Roy F. and Norman A. Chandler. White Feather: Carlos Hathcock, USMC Scout Sniper. New York, New York: Iron Brigade Publishing, 1997.
Henderson, Charles. Marine Sniper: 93 Confirmed Kills. New York, New York: The Berkely Publishing Group, 1986.
To this day, Carlos Hathcock continues to remain one of the greatest snipers of all time. Even the late Chris Kyle – a legend in his own right – acknowledged the greatness of Hathcock and his work as a sniper. Hathcock's dedication and commitment to duty helped to save countless American lives during the war in Vietnam, and continues to inspire snipers and servicemen today (worldwide). Although his life was tragically cut short (from complications arising from multiple sclerosis), his legend continues to live on in the hearts and minds of others.
Works Cited:
Images / Photographs:
Stillwell, Blake. "This Marine Was The 'American Sniper' Of The Vietnam War." Military.com. Accessed March 21, 2019. https://www.military.com/marine-corps-birthday/carlos-hathcock-famous-marine-corps-sniper.html.
Wikipedia contributors, "Carlos Hathcock," Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Carlos_Hathcock&oldid=884167496 (accessed March 21, 2019).
This content is accurate and true to the best of the author's knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
© 2019 Larry Slawson
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Eric Dierker
10 months ago from Spring Valley, CA. U.S.A.
You are a great history teacher. Thank you. Never heard of him before.
Jeff Zod
10 months ago from Nairobi
I am so glad you wrote it. I really enjoyed it. You have outdone yourself as usual.
10 months ago from North Carolina
@Jeff I'm so glad you enjoyed! Been wanting to do an article on Carlos Hathcock for quite some time. He truly is the greatest sniper of all time. I recently read a biography on him, and it was absolutely fascinating. He did some amazing things.
I love this article.Snipers have always fascinated me.Carlos Hathcock is one of the greatest snipers all time.


text:
Q: Match Row based on previously available date to second table I have two tables of data I would like to join: Transactions and User History
Transactions

user_id
transaction_id
price
transaction_date
redeemed_points

12345
rztgdh
0.00
2022-10-07 10:25:43
1500

54321
lfhskfdj
0.00
2022-10-11 17:52:16
2100

User History

user_id
time_updated
point_balance

12345
2022-09-26 18:15:38
3160

12345
2022-09-29 13:19:22
3250

12345
2022-10-07 10:25:45
1750

54321
2022-10-05 19:43:29
5000

54321
2022-10-11 17:52:18
2900

The transactions table should be the left join table and I want to add a column to it with the point balance prior to the start of the transaction the most up to date previous timestamp. Meaning the first transaction from user 12345 should have an additional column with the last available point balance from the user history table prior to the transaction, in this case 3250.
The final table should look like this
Transactions with new column

user_id
transaction_id
price
transaction_date
redeemed_points
points_start

12345
rztgdh
0.00
2022-10-07 10:25:43
1500
3250

54321
lfhskfdj
0.00
2022-10-11 17:52:16
2100
5000

Any help is much appreciated.
I have been unable to come up with a solution that would be close to what I want to achieve and be automated enough. I am working with datasets of about 200k rows, any manual work is not possible.

A: Use SUMIFS with a nested AGGREGATE:
=SUMIFS(J:J,H:H,A2,I:I,AGGREGATE(14,7,$I$2:$I$6/(($H$2:$H$6=A2)*($I$2:$I$6<D2)),1))

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The Pets Play Squeaking Dice Dog Toys are available in two packs with assorted colours.
The Pets Play Squeaking Dice Dog Toys are brilliant items for dog owners. Made from a durable and save material, dogs and young puppies can chew and play with the Pets Play Squeaking Dice Dog Toys at home, in the park or garden.