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?