Bisimplicial sets and homology I'm not sure about the following result:
Theorem (?): Let $f_{\bullet,\bullet}: X_{\bullet,\bullet}\rightarrow Y_{\bullet,\bullet}$ a map of bisimplicial sets such that for any 
natural number $n$,  $H_{\ast}(X_{\bullet,n},\mathbb{Z})\rightarrow H_{\ast}(Y_{\bullet,n},\mathbb{Z})$is an isomorphism, then
$H_{\ast}(DX_{\bullet,\bullet},\mathbb{Z})\rightarrow H_{\ast}(DY_{\bullet,\bullet},\mathbb{Z})$ is an isomorphism.
Where $DX_{\bullet,\bullet}$ is the diagonal simplicial set.
I was wondering about the following proposition: 
Let $f_{\bullet,\bullet}: X_{\bullet,\bullet}\rightarrow Y_{\bullet,\bullet}$ a map of bisimplicial sets such that for any 
natural number n, $H_{\ast}(X_{\bullet,n},\mathbb{Z})\rightarrow H_{\ast}(Y_{\bullet,n},\mathbb{Z})$ is surjective (resp. injective), then
$H_{\ast}(DX_{\bullet,\bullet},\mathbb{Z})\rightarrow H_{\ast}(DY_{\bullet,\bullet},\mathbb{Z})$ is surjective (resp. injective).
Since I don't know how to prove the first theorem (?), I do not know if the second proposition has any chance to be correct? Are there any similar results in the literature?
 A: Yes, this is true. I will suppress $\mathbb{Z}$ from the notation, this works with any coefficients or indeed with any homology theory.
We have isomorphisms $H_*(B, A) \cong \tilde H_*(B/A) \cong \tilde H_{*+k}(\Sigma^k B/A) \cong H_{*+k}(B \times \Delta[k], A \times \Delta[k] \cup B \times \partial\Delta[k])$. Thus if $A \hookrightarrow B$ is a homology equivalence, then so is $A \times \Delta[k] \cup B \times \partial\Delta[k]\hookrightarrow B \times \Delta[k]$.
From here you can use any of the two standard arguments for weak homotopy equivalences.
Goerss, Jardine (Proposition IV.1.9) provide an explicit filtration of the diagonal with each step obtained from the previous one by pushing out a pushout product map as above.
Alternatively, Hovey's proof (Lemma 5.3.1) also applies verbatim. It shows that the diagonal functor is left Quillen with respect to the Reedy model structure (starting with the homological Bousfield localization of course). Indeed, its right adjoint is $K \mapsto \mathsf{sSet}(\Delta[-],K)$ and it is right Quillen by the adjoint form of the pushout product property above (sometimes called the pullback hom property).
