$\newcommand{\V}{\mathscr{V}}\newcommand{\A}{\mathcal{A}}\newcommand{\B}{\mathcal{B}}\newcommand{\C}{\mathcal{C}}$Fix a closed symmetric monoidal category $\V$, writing the product as $\otimes$, the hom-objects as $\underline{\V}(-,-)$ and the monoidal unit as $\ast$.
In my "research" I encountered and solved the following problem:
If $\underline{\A},\underline{\B}$ are $\V$-categories then find a suitable concept of a $\V$-category $\underline{\A\times\B}$ generalising certain phenomena from ordinary category theory
For me, the phenomena were related to adjunctions with parameter and lifting intertwined families of functors $\A\to\C,\B\to\C$ to some kind of combined bifunctor object $\A\times\B\to\C$. This is probably extremely standard, but I wouldn't know.
I define $\underline{\A\times\B}$ to have objects pairs $(\alpha,\beta)$ for $\alpha\in\A,\beta\in\B$ and hom-objects shall be: $$\underline{\A\times\B}((\alpha,\beta),(\alpha',\beta')):=\underline{\A}(\alpha,\alpha')\otimes\underline{\B}(\beta,\beta')$$ Together with obvious definitions of identity arrow and composition. This really does form a $\V$-category, and it served the purposes I needed it to.
My question is, are there situations where we can make sense of this category as a genuine product object? A priori I think there need not be any enriched projection functor $\underline{\A\times\B}\to\underline{\A}$, because that demands a way of defining arrows: $$\underline{\A}(\alpha,\alpha')\otimes\underline{\B}(\beta,\beta')\overset{??}{\longrightarrow}\underline{\A}(\alpha,\alpha')$$ Which I don't see how to do. Moreover, it's not clear to me what the underlying category $\underline{\A\times\B}_0$ should be: it should in some sense "contain" the product of underlying categories $(\underline{\A})_0\times(\underline{\B})_0$ since for a pair of arrows $f:\alpha\to\alpha',g:\beta\to\beta'$ in the underlying categories I have an arrow: $$\ast\cong\ast\otimes\ast\overset{\overline{f}\otimes\overline{g}}{\longrightarrow}\underline{\A}(\alpha,\alpha')\otimes\underline{\B}(\beta,\beta')$$Which defines an arrow in $\underline{\A\times\B}_0$. But to the converse, I don't see how we can characterise generic arrows: $$\ast\overset{?}{\longrightarrow}\underline{\A\times\B}(-,-)$$So there could conceivably be more arrows in $\underline{\A\times\B}_0$, and these additional arrows are currently mysterious to me.
I am motivated to ask this question by the fact that if $\V=\mathsf{Set}$ - or more generally, if $\V$ is a Cartesian closed category with $\times$ as its monoidal product - then it is clear that $\underline{\A\times\B}$ is the same thing as the ordinary product of ordinary categories. So, what conditions can one impose on $\V$ to understand $\underline{\A\times\B}$ as a product? Or at least, to understand what the underlying category of $\underline{\A\times\B}$ looks like? As it stands, I can note only one interesting property of $\underline{\A\times\B}$:
If $\underline{\C}$ is a $\V$-category and for each $(\alpha,\beta)\in\A\times\B$ there are $\V$-functors $F^\alpha:\underline{\B}\to\underline{\C},F^\beta:\underline{\A}\to\underline{\C}$ satisfying suitable a 'intertwining' relationship, we can uniquely lift these data to a $\V$-functor $F:\underline{\A\times\B}\to\underline{\C}$.
Are there any more?
