Corollary 4.3.7.3. Let $f: K \rightarrow X$ and $q: X \rightarrow S$ be morphisms of simplicial sets. Then:
If $q$ is a left fibration, then the induced map
\[ X_{/f} \rightarrow X \times _{ S} S_{/ (q \circ f)} \]is a Kan fibration.
If $q$ is a right fibration, then the induced map
\[ X_{f/} \rightarrow X \times _{ S} S_{(q \circ f)/} \]is a Kan fibration.