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 $\mathcal{M}$?