Assume given a pullback square of simplicial categories


$$\begin{array}[c]{ccc}
A&{\rightarrow}&B\\
\downarrow&&\downarrow\\
C&{\rightarrow}&D.
\end{array}$$


Suppose further that one of the induced arrows $Ho (B) \to Ho(D)$ or $Ho(C) \to Ho(D)$ is an isofibration, and for each couple of objects $x,y \in A$, the induced pullback square of simplicial mapping spaces (I abuse the notation by writing $x,y$ instead of their images in $B,C,D$)
$$\begin{array}[c]{ccc}
A(x,y)&{\rightarrow}&B(x,y)\\
\downarrow&&\downarrow\\
C(x,y)&{\rightarrow}&D(x,y)
\end{array}$$


is a homotopy pullback.


Does this imply that the original square is in fact a homotopy pullback of simplicial categories? 



Criterion for homotopy pullback square of simplicial categories
Yes. In fact such a square can be replaced with a weakly equivalent Reedy fibrant pullback square without changing the object set of any of the simplicial categories. For a proof see, e.g., Lemma 3.1.11 in [this paper](https://arxiv.org/abs/1612.02608).


