Let $Set$ be the category of sets and $sSet$ the category of simplicial sets (the category of functors from $\mathbb{\Delta}^{op}$ to $Set$). Every functor $F:Set \to Set$ induces a functor $\hat{F}:sSet\to sSet$ by post-composition with $F$.
Question: Is it always true that $F$ preserves weak homotopy equivalences of simplicial sets?
I can't find a reason why it should or an example that it doesn't. The question is just out of curiosity, but I would also like to understand how does one approach such a question.
One particular functor I am thinking about is the power set functor $P$ (as a covariant functor) that takes every set to the set of its subsets and acts on morphisms by taking direct image of subsets. Does $\hat{P}$ preserve weak equivalences? If so, does it have a known homotopical meaning?