Variant 4.4.5.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset which contains every vertex of $B$. Then the induced map
\[ F': \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}}) \]
is an isofibration of $\infty $-categories.