Will a homotopical right-adjoint preserve homotopy limits?

Question

Let $M$ and $N$ be homotopical categories (both of them complete) and $I$ a small category. Assume that we have right deformations for the limit functors $M^I \to M$ and $N^I \to N$, so we have the (non-total) right derivatives \begin{align} \text{holim} : M^I &\to M \\ \text{holim} : N^I &\to N.\end{align} Assume now that we have a right-adjoint functor $F: M \to N$ (so it commutes with the limit) that is additionally homotopical, i.e. it preserves all weak equivalences. Question: Is there a natural weak equivalence between $\text{holim} \circ F$ and $F\circ \text{holim}$? In other words: Will $F$ (applied to diagrams) also preserve homotopy limits? If this is not true, what might be additional requirements on $F$ to make it true?

Thank you for any hints.

Addendum: I have a little more concrete problem in mind, in which $M$ is given by non-negative cochain complexes and $N$ by non-negative chain complexes. I think I could perform a proof by direct computation if I can find concrete descriptions of the cotensors in these simplicial model categories. I only found them for (co)simplicial Abelian groups and I am a little unsure how to transfer them to complexes. Has anyone seen that written down explicitly?

Answer 1

The tensors and cotensors in chain complexes come via the Moore complex realization of simplicial sets. For instance, the Moore complex of the interval over a commutative ring $R$ is $...\to 0\to R\to R\oplus R\to 0\to...$ where the nontrivial map is $(1,-1)$. The cotensor with the interval, that is, the path object, is the complex of maps out of this complex. In general, you can write down a formula for any cotensor as a limit of cotensors by simplices, although that might not be explicit enough for your needs.

For your abstract problem: let the deformations be $R_M, R_N$. Then $$F(\mathrm{holim}D)=F( \mathrm{lim}(R_M(D)))\cong \mathrm{lim}F\circ R_M(D)...\mathrm{lim}R_N F\circ D=\mathrm{holim} F\circ D$$ So we see the problem is concentrated in that ellipsis. We can fill it in using the weak equivalences $r_D:D\to R_MD,r_{F\circ D}:F\circ D\to R_N(F\circ D)$, but only as a zigzag.