Let $M$ and $N$ two very nice simplicial model category and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow Ho(M)$ induces an equivalence of homotopy categories and it is well defined. We define the category $\pi_{0}M$ as the category having the same objects of $M$ and $Hom_{\pi_{0}M}(a,b)=\pi_{0}Map_{M}(a,b)$.

What can we say about the functor $\pi_{0}F:\pi_{0}M\rightarrow \pi_{0}N$? is it an equivalence of categories. Maybe we should assume that $F$ takes cofibrant-fibrant objects to cofibrant-fibrant objects... or something close to that.