The problem comes from a paper of Bertrand Toën, Homotopical Algebraic Geometry II, Appendix A, Prop A.0.3.
Let $M $ be a model category and $C $ be a full subcategory of the category of weak equivalences in $M$. Denote by $SSet$ the category of simplicial sets and equip it with the usual model structure (cf. Mark Hovey’s book Model Categories).
Let $F:C^{op}\rightarrow SSet$ be a functor sending morphisms of $C$ to weak equivalences in $SSet$ and such that $F(x)$ is a fibrant object in $SSet$ for any object $x\in C$. Let $N(F)$ be the simplicial set defined as follows:
Let $F_n$ be the functor $C^{op}\rightarrow Sets$ sending $x$ to $F(x)_n$. Denote by $C/F_n$ the category of elements of $F_n$, i.e. its objects are pairs $(c,\alpha \in F_n(c))$ and morphisms are the obvious one. Then $N(F)$ is the diagonal of the bisimplicial set $$(n,m)\rightarrow N(C/F_n)_m$$ There is an obvious map $N(F)\rightarrow N(C)$ where the latter is the nerve of the category $C$.
Under these assumptions, Toën says there is a standard lemma, which shows that the homotopy fiber at $x\in C$ of $N(F)\rightarrow N(C)$ is naturally equivalent to $F(x)$.
My attempt: it seems that $F(x)$ is the fiber of the diagram. What’s next to do? Any comment will be appreciated.