Corollary 4.5.5.16. Let $q: X \rightarrow S$ be an isofibration of simplicial sets. Suppose we are given a morphism of simplicial sets $B \rightarrow S$ and a simplicial subset $A \subseteq B$. Then the restriction map $\theta : \operatorname{Fun}_{/S}( B, X ) \rightarrow \operatorname{Fun}_{/S}(A, X)$ is an isofibration of $\infty $-categories.
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Proof. The morphism $\theta $ is a pullback of the isofibration $\operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S)$ of Proposition 4.5.5.14, and is therefore also an isofibration (Remark 4.5.5.11). We conclude by observing that since $q$ is an inner fibration (Remark 4.5.5.7), the simplicial sets $\operatorname{Fun}_{/S}(B,X)$ and $\operatorname{Fun}_{/S}(A,X)$ are $\infty $-categories (Proposition 4.1.4.6). $\square$