If $C$ is a 1-category and $D$ is a locally presentable $(\infty,1)$-category presented by a combinatorial model (1-)category $M$, then any $(\infty,1)$-functor $C\to D$ can be represented by a strict 1-functor $C\to M$, because the functor category $M^C$ carries a model structure that presents the $(\infty,1)$-functor category $D^C$.
I would like to know if something analogous is true when all the 1s are replaced by 2. In particular, I care about the situation where $D = \rm Cat_{(\infty,1)}$ is the $(\infty,2)$-category of $(\infty,1)$-categories and $M = \rm sCat$ is the model category of simplicially enriched categories, which is a 2-category in an evident way with simplicial natural transformations. That is, if $C$ is a 2-category, can an $(\infty,2)$-functor $C \to \rm Cat_{(\infty,1)}$ be represented by a 2-functor, or at least a pseudofunctor, $C \to \rm sCat$?
