Is $Cat_{\Delta}$ enriched over itself?

Question

Question is just as in the title. Is the category of simplicially enriched categories enriched over itself? If not, is it enriched over another relevant category, e.g., simplicial sets?

Answer 1

It is possible to have an "enriched functor category" when we consider $Set_{\Delta}$-enriched functors $F,G : \mathscr{C} \rightarrow \mathscr{M}$ where $\mathscr{M}$ is a simplicial model category. Define a morphism $\eta : F \rightarrow G$ as a collection of maps in $\mathscr{M}$, $\eta_{x}: F(x) \rightarrow G(x)$, such that the diagram

$\require{AMScd}$ \begin{CD} \mathscr{C}(x,y) \otimes F(x) @>>> F(y)\\ @VVV @VVV\\ \mathscr{C}(x,y) \otimes G(y) @>>> G(y) \end{CD}

commutes in $\mathscr{M}$. The category with objects enriched functors and morphisms the natural transformations described above is a simplicially enriched category. The reason this works is precisely because $\mathscr{M}$ is powered and tensored over $Set_{\Delta}$. I do not think this can be made to work in general, unless we add hypotheses on the codomain $Set_{\Delta}$-category.