Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a simplicial category (though this enrichment does not cooperate with the Bergner model structure, for instance). I can apply the coherent nerve functor and produce two quasicategories $N(C)$ and $N(D)$ which have a Kan complex of morphisms between them $Map(N(C),N(D))$. I'm interested in knowing if there's any way to control $Map(N(C),N(D))$ (even just up to homotopy) if I know what $sFun(C,D)$ is.

I could control the homotopy type of $Map(N(C),N(D))$ if I knew, for instance, that $C$ was also cofibrant, but this seems to be an extremely strong condition to require. In general, is there just no useful way to get information about $Map(N(C),N(D))$ from $sFun(C,D)$?

I should also add that in the example I'm most interested in, $C$ is just a discrete category (thought of as a simplicial category), so maybe that gives me extra control over its cofibrant replacement?

spaceof functors $C \to D$, or the$\infty$-categoryof functors $C \to D$? If I understand the notation $sFun(C,D)$ correctly, it will in general have non-invertible morphisms and be more closely related to the latter -- you'd have to throw away non-invertible morphisms to get something like the former. $\endgroup$ – Tim Campion♦ Sep 25 '20 at 21:31