Kerodon

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

Corollary 5.6.4.7. Suppose we are given a commutative diagram of simplicial sets

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

where $U$ and $V$ are cocartesian fibrations. Then $F$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ (Definition 5.1.6.1) if and only if it is a categorical equivalence of simplicial sets.

Proof. Combine Proposition 5.1.6.5 with Corollary 5.6.4.5. $\square$