2
$\begingroup$

Let $\mathscr C,\mathscr D$ be (right) closed monoidal categories. Then both of them can be considered as enriched over themselves via their internal homs, which I will denote by $\textbf{Maps}$. Now if $F\colon\mathscr C\to\mathscr D$ is any lax monoidal functor, we have a $\mathscr D$-enriched category $F_*\mathscr C$ with the same objects as $\mathscr C$ and morphism objects $\textbf{Maps}_{F_*\mathscr C}(X,Y)=F(\textbf{Maps}_{\mathscr C}(X,Y))$; composition and unit are induced from the ones in $\mathscr C$ using $F$ and its structure maps.

Everything of the above can of course be found in the classic references. But I'm interested in slightly more: we have for any $X,Y\in\mathscr C$ a canonical map $\textbf{Maps}_{F_*\mathscr C}(X,Y)\to \textbf{Maps}_{\mathscr D}(FX,FY)$ in $\mathscr D$, given as the canonical mate of the structure map $\psi$ of $F$, i.e. as the composition $$F(\textbf{Maps}_{\mathscr C}(X,Y))\xrightarrow{\eta}\textbf{Maps}_{\mathscr D}(F(X), F(\textbf{Maps}_{\mathscr C}(X,Y))\otimes F(X))\xrightarrow{\psi}\textbf{Maps}_{\mathscr D}(F(X), F(\textbf{Maps}_{\mathscr C}(X,Y)\otimes X))\xrightarrow{\epsilon}\textbf{Maps}_{\mathscr D}(F(X), F(Y))$$ where we use the unit of the tensor-hom adjunction in $\mathscr D$ and the counit of the one in $\mathscr C$. I'm looking for a reference that proves that these maps assemble into a $\mathscr D$-enriched functor $F_*{\mathscr C}\to\mathscr D$.

I have checked this for myself, but this seems so basic that I would be surprised if it weren't written up somewhere. So, does anyone know a reference for this? (In the cases I'm interested in, the monoidal structures on $\mathscr C$ respectively $\mathscr D$ are actually the cartesian ones, so a reference for this special case would be perfectly fine.)

Thanks in advance!

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.