In particular I'm interested in this situation. Let $sSet^{2}$ denote the category of bisimplicial sets with diagonal model structure (weak equivalences are diagonal weak equivalences and cofibrations are monomorphisms) and $sSet$ be the category of simplicial sets with usual model structure. Then we know $d$ (the diagonal functor) is a right Quillen functor. Now following is the proof of the claim $\forall X \in {sSet^2}^I $ $$ d(holimX)\simeq (holim (dX)).$$

We denote the induced diagonal functor on diagram category (which is again a right Quillen functor) by $d$ as well.

Proof: Let $X'$ be fibrant replacement of $X$ in the injective model structure on ${sSet^2}^I$. Hence the homotopy limit $holimX$ is weakly equivalent to limit $limX'$ . Since this is a diagonal weak equivalence we have, $$ d(holimX)\simeq d(limX').$$ Furthermore, $d$ is a right adjoint, hence the R.H.S. above is weakly equivalent to $lim(dX')$. Moreover $d$ is right Quillen functor hence it preserves fibrant objects, so $dX'$ is fibrant. Again the limit $lim(dX')$ is weakly equivalent to the homotopy limit $holim(dX)$.

Is this proof correct?