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?