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Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow Ho(M)$ is 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 $\pi_{0}F:\pi_{0}N\rightarrow \pi_{0}M$? 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.

Clarification: A simplicial functor between simplicial model categories $M$ and $N$ is a simplicial enriched functor between simplicial enriched categories.

Clarification II: A simplicial model category is a model category $M$ tensored and cotensored over the model category of simplicial sets in a way compatible with model structure of both model structure on $M$ and on simplicial sets.

Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow Ho(M)$ is 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 $\pi_{0}F:\pi_{0}N\rightarrow \pi_{0}M$? 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.

Clarification: A simplicial functor between simplicial model categories $M$ and $N$ is a simplicial enriched functor between simplicial enriched categories.

Clarification II: A simplicial model category is a model category $M$ tensored and cotensored over the model category of simplicial sets in a way compatible with model structure of both model structure on $M$ and on simplicial sets.

Edit I think there is a counterexample: Let $M=N$ be the standard model category of simplicial sets, and let $F: sSet\rightarrow sSet$ be the functor of fibrant replacement such that it is simpllicial (I guess such functor do exist). Then $\pi_{0}F$ is not an equivalence of categories, while $Ho(F)$ is. I'm I right ?

Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow Ho(M)$ is 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 $\pi_{0}F:\pi_{0}N\rightarrow \pi_{0}M$? 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.

Clarification: A simplicial functor between simplicial model categories $M$ and $N$ is a simplicial enriched functor between simplicial enriched categories.

Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow Ho(M)$ is 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 $\pi_{0}F:\pi_{0}N\rightarrow \pi_{0}M$? 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.

Clarification: A simplicial functor between simplicial model categories $M$ and $N$ is a simplicial enriched functor between simplicial enriched categories.

Clarification II: A simplicial model category is a model category $M$ tensored and cotensored over the model category of simplicial sets in a way compatible with model structure of both model structure on $M$ and on simplicial sets.

Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow Ho(M)$ is 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 $\pi_{0}F:\pi_{0}N\rightarrow \pi_{0}M$? 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.

Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow Ho(M)$ is 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 $\pi_{0}F:\pi_{0}N\rightarrow \pi_{0}M$? 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.

Clarification: A simplicial functor between simplicial model categories $M$ and $N$ is a simplicial enriched functor between simplicial enriched categories.