Corollary 4.5.2.34. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an isofibration of $\infty $-categories, let $B \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $f: A \rightarrow B$ be a categorical equivalence of simplicial sets. Then precomposition with $f$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( B, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{C}}}(A,\operatorname{\mathcal{E}})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Corollary 4.5.2.32 to the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B, \operatorname{\mathcal{E}}) \ar [r]^-{ \circ f} \ar [d]^{U \circ } & \operatorname{Fun}(A, \operatorname{\mathcal{E}}) \ar [d]^{U \circ } \\ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \ar [r]^-{\circ f} & \operatorname{Fun}(A, \operatorname{\mathcal{C}}); } \]
note that the vertical maps are isofibrations (Corollary 4.4.5.6) and the horizontal maps are equivalences of $\infty $-categories (Proposition 4.5.3.8). $\square$