Let $\mathcal V$ be a closed symmetric monoidal model category. It is well known that the category $\mathcal V$-$\mathbf{Cat}$ of $\mathcal V$-enriched categories is itself a closed symmetric monoidal category. The paper by Berger-Moerdijk shows that in some cases $\mathcal V$-$\mathbf{Cat}$ admits the so-called *canonical model structure* (weak equivalences are Dwyer-Kan equivalences). Typical examples are obtained taking $\mathcal V = \mathbf{Set}$, $\mathcal V = \mathbf{sSet}$ or $\mathcal V = \mathbf{Ch}(k)$ (the category of complexes of $k$-modules). My question is the following:

Are there reasonable hypothesis on $\mathcal V$ which ensure that $\mathcal V$-$\mathbf{Cat}$ is a closed symmetric monoidal model category?

The case $\mathcal V = \mathbf{Ch}(k)$ gives the known model structure on the category $\mathbf{dgCat}$ of dg-categories, and the homotopy category $\mathrm{Ho}(\mathbf{dgCat})$ is indeed a closed symmetric monoidal category - even if I actually don't know if $\mathbf{dgCat}$ is a monoidal model category (with the right compatibility conditions). Is a similar result reasonable also for $\mathcal V = \mathbf{sSet}$? I couldn't find any reference about internal homs in the homotopy category of simplicially enriched categories, even if I have in mind some possible candidates...

PS: this question is possibly related to this one.

anynontrivial monoidal model categories $\mathcal{V}$ for which the standard monoidal structure on $\mathcal{V}\text{-}\mathbf{Cat}$ makes it a monoidal model category. (Of course, it is true in trivial cases like $\mathcal{V}=\mathbf{Set}$.) The closest thing I know of is that for $\mathcal{V}=\mathbf{Cat}$, there is a different monoidal structure on $2\text{-}\mathbf{Cat}$ called the Gray tensor product that makes it a monoidal model category (web.science.mq.edu.au/~slack/papers/qmc2cat.html). $\endgroup$ – Mike Shulman Dec 24 '18 at 11:48