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?
