Corollary 4.3.7.13. Let $q: X \rightarrow S$ be a morphism of simplicial sets, and let $x \in X$ be a vertex having image $s = q(x)$ in $S$. Then:

If $q$ is a right fibration, then the induced map $X_{/x} \rightarrow S_{/s}$ is a trivial Kan fibration.

If $q$ is a left fibration, then the induced map $X_{x/} \rightarrow S_{s/}$ is a trivial Kan fibration.