Corollary 4.4.5.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. Suppose we are given a morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{D}}$ and a simplicial subset $A \subseteq B$. Then the restriction map $\theta : \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( A, \operatorname{\mathcal{C}})$ is an isofibration of $\infty $-categories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $V$ be the collection of all vertices of $B$ which are not contained in $A$, which we regard as a (discrete) simplicial subset of $B$. Then $\theta $ factors as a composition
\begin{eqnarray*} \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}(B, \operatorname{\mathcal{C}}) & \xrightarrow {\theta '} & \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( A \cup V, \operatorname{\mathcal{C}}) \\ & \simeq & \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( A, \operatorname{\mathcal{C}}) \times \prod _{v \in V} \operatorname{\mathcal{C}}_{v} \\ & \rightarrow & \operatorname{Fun}_{ /\operatorname{\mathcal{D}}}(A, \operatorname{\mathcal{C}}), \end{eqnarray*}
where the last map is an isofibration by virtue of the fact that each $\operatorname{\mathcal{C}}_{v} = \{ v\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is an $\infty $-category. It will therefore suffice to show that $\theta '$ is an isofibration. This follows from Variant 4.4.5.11, since $\theta '$ is a pullback of the restriction map
\[ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( A \cup V, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A \cup V, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}}). \]
$\square$