$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 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.

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, and is therefore also an isofibration (Remark We conclude by observing that since $q$ is an inner fibration (Remark, the simplicial sets $\operatorname{Fun}_{/S}(B,X)$ and $\operatorname{Fun}_{/S}(A,X)$ are $\infty $-categories (Proposition $\square$