Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 5.1.6.15. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr]_{U} & & \operatorname{\mathcal{D}}\ar [dl]^{V} \\ & \operatorname{\mathcal{E}}, & } \]

where $U$ and $V$ are right fibrations. Then $F$ is an equivalence of inner fibrations if and only if, for every vertex $E \in \operatorname{\mathcal{E}}$, the induced map $F_{E}: \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \{ E \} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is a homotopy equivalence of Kan complexes.