I'm just reformulating your question in simplicial case.
If you consider the category of simplicial sets $\mathbf{sSet}$ you can formulate your question as follows:
Is the diagonal functor $diag: [\mathbf{\Delta}^{op},\mathbf{sSet}]\rightarrow \mathbf{sSet}$ from bisimplicial sets to simplicial sets homotopicaly full ?
The diagonal functor is actually the realization functor. If we put the diagonal model structure on $ [\mathbf{\Delta}^{op},\mathbf{sSet}]$ then $diag$ is a Quillen equivalence.
As a consequence, if $f: diag(X_{\bullet, \bullet})\rightarrow diag(Y_{\bullet, \bullet})$
is a bisimplicial map and $X_{\bullet, \bullet}$, $Y_{\bullet, \bullet}$ are fibrant-cofibrant bisimplicial sets (in the diagonal model structure), then there exists $g: X_{\bullet, \bullet}\rightarrow Y_{\bullet, \bullet}$ such that $diag(g)$ is homotopic to $f$.