Given two bisimplicial sets $X_{\bullet,\bullet}$ and $Y_{\bullet,\bullet}$, we have the result that if given a map $f:X_{\bullet,\bullet}\to Y_{\bullet,\bullet}$ such that the restriction $f_n:X_{n,\bullet}\to Y_{n,\bullet}$ is a homotopy equivalence for each n then the induced map on the diagonal simplicial sets $d(f):d(X)_{\bullet}\to d(Y)_{\bullet}$is also a homotopy equivalence.
Is the converse of this statement true? If not what is a counterexample?
The converse is false. After all, a lot of information is lost when taking the diagonal.
Explicitly: any simplicial set can be considered as a degreewise discrete bisimplicial set; under this correspondence, if $f : X \to Y$ is a weak homotopy equivalence of simplicial sets, then $f_\bullet : X_\bullet \to Y_\bullet$ is a morphism of bisimplicial sets whose diagonal is a weak homotopy equivalence, but it is a degreewise weak homotopy equivalence if and only if $f : X \to Y$ is an isomorphism.