Kerodon

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

Proof. For every pair of $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, the core $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ is a Kan complex (Corollary 4.4.3.11). It follows that the simplicial category $\operatorname{QCat}$ of Construction 5.4.4.1 is locally Kan, so its homotopy coherent nerve $\operatorname{\mathcal{QC}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$ is an $\infty $-category by virtue of Theorem 2.4.5.1. $\square$