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Corollary Let $q: X \rightarrow S$ be an inner fibration of simplicial sets and let $f: K \rightarrow X$ be any morphism of simplicial sets. Then the restriction map

\[ X_{/f} \rightarrow X \times _{S} S_{ / (q \circ f)} \]

is a right fibration, and the restriction map

\[ X_{f/} \rightarrow X \times _{ S } S_{(q \circ f)/} \]

is a left fibration.

Proof. Apply Proposition in the special case $K_0 = \emptyset $. $\square$