Kerodon

I don't quite follow the argument here. I think there are some references missing.

In the case where the projection $q\colon X\times X\to X$ to the first factor is a Kan fibration between Kan complexes it is asserted that $q$ is a weak homotopy equivalence if and only if the fibers $q^{-1}\lbrace x\rbrace$ are contractible Kan complexes. Corollary 3.2.7.4 gives that the latter holds if and only if $q$ is a homotopy equivalence. But why is this equivalent to $q$ being a weak homotopy equivalence?

Similarly, why is contractibility of $X$ (provided $X$ is a non-empty Kan complex) equivalent to weak contractibility?

A map of Kan complexes is a homotopy equivalence if and only if it is a weak homotopy equivalence.