Corollary 4.5.2.35. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories, let $A \subseteq B$ be simplicial sets, and suppose we are given a diagram $A \rightarrow \operatorname{\mathcal{C}}$. Then postcomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{A/}(B, \operatorname{\mathcal{D}})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Corollary 4.5.2.32 to the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \ar [r]^-{ F \circ } \ar [d] & \operatorname{Fun}(B, \operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \ar [r]^-{F \circ } & \operatorname{Fun}(A, \operatorname{\mathcal{D}}); } \]
note that the vertical maps are isofibrations by virtue of Corollary 4.4.5.3 and the horizontal maps are equivalences by virtue of Remark 4.5.1.16. $\square$