I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this?
The second one is much more open, it's basically what restricitions do we have to put on a cosimplicial object $U:\Delta \rightarrow C$ such that the functor $$F_U:C \rightarrow \text{s}\textbf{Set}$$ given by $$F_U(c)_n = \operatorname{Hom}(U_n,c)$$ preserves weak equivalences and/or cofibrations between sufficiently nice objects?