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Proposition 4.6.1.22. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset which contains every vertex of $B$, and let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the induced map

\[ \operatorname{Fun}( B, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ f\} \rightarrow \operatorname{Fun}( B, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \{ q \circ f \} \]

is a Kan fibration of simplicial sets.

Proof. It follows from Proposition 4.6.1.9 that the simplicial sets $\operatorname{Fun}( B, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ f\} $ and $\operatorname{Fun}( B, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \{ q \circ f \} $ are Kan complexes. It will therefore suffice to show that $\theta $ is an isofibration (Corollary 4.4.3.10). This follows from the observation that $\theta $ is a pullback of the restriction map

\[ \operatorname{Fun}( B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( B, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}( A, \operatorname{\mathcal{C}}), \]

which is an isofibration by virtue of Variant 4.4.5.11. $\square$