How to define the internal hom between presheaves valued in cotensored categories?

Question

First let $\mathcal{V}$ be a closed symmetric monoidal category and $\mathcal{M}$ be a category enriched over $\mathcal{V}$. Moreover we assume $\mathcal{M}$ is cotensored, or powered over $\mathcal{V}$. This implies that for any object $v\in \mathcal{V}$ and $m\in \mathcal{M}$ we have an object $m^v\in \mathcal{M}$ and we can think of it as the internal hom from $v$ to $m$. An example is to take $\mathcal{V}=Ch_k$ the category of chain complexes over a field $k$ and $\mathcal{M}$ be a dg-category over $k$.

Now let $X$ be a scheme and $\mathcal{S}$ be the site of open covers of $X$. Let $F$ and $G$ be two presheaves: $F: \mathcal{S}^{op}\to \mathcal{V}$ and $G: \mathcal{S}^{op}\to \mathcal{M}$. Can we define the internal hom $G^F$ to make it an object of $\mathcal{M}$?

Answer 1

Yes, if $\mathcal M$ is reasonable. The object-wise internal hom $(s,t) \mapsto G(s)^{F(t)}$ is contravariant in $s \in \mathcal S$ and covariant in $t$. The object you want is the end of this functor, i.e. the maximal subobject of $\prod_{s\in \mathcal S} G(s)^{F(s)}$ satisfying that if you project to the $u$th component and then apply $F(f:u\to v)$ to land in $G(u)^{F(v)}$, that's the same as projecting to the $v$th component and then applying $G(f:u\to v)$ to land in $G(u)^{F(v)}$. This exists provided $\mathcal M$ has sufficient limits.