Kerodon

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

Corollary 5.1.6.8. 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 inner fibrations and $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex. Then $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$ if and only if it an equivalence of $\infty $-categories.

Proof. Our assumption that $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex guarantees that the inner fibrations $U$ and $V$ are isofibrations (Example 4.4.1.6), so the desired result follows from Proposition 5.1.6.5. $\square$