Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category 
$\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $B:\mathbf{Cat}\rightarrow\mathbf{Top}$ be the classifying space functor (take nerve then realize). How do
$B\underline{\mathbf{sCat}}(\mathcal{C},\mathcal{D})$ and $\underline{\mathbf{sTop}}(B\mathcal{C},B\mathcal{D})$ compare? 


I think they are weakly equivalent (in the Reedy model structure), and I'm hoping that there might be a trivial cofibration between them. Anyone know a reference for something like this? 



Compatibility of classifying space with inner-hom?
Here's something easier. Let $C$ and $D$ be categories, and ask: is there an equivalence between $B\underline{\mathrm{Cat}}(C,D)$ (the classifying space of the functor category) and $\underline{\mathrm{Top}}(BC,BD)$ (the mapping space between two classifying spaces)?


In general, the answer is no. For instance, let $C$ be the category $(x\rightrightarrows y)$ with two objects and two non-identity arrows from $x$ to $y$. Then $BC$ is a circle, and $\underline{\mathrm{Top}}(BC,BC)$ is homotopy equivalent to $\mathbb{Z}\times S^1$, whereas $B\underline{\mathrm{Cat}}(C,C)$ is not (it has only finitely many components, for instance). 


Taking constant simplicial objects should provide a counterexample to your claim, unless I've misunderstood what you are asking for.


