Proposition 8.6.7.5. The simplicial set $\operatorname{\mathcal{QC}}^{\asymp }$ is an $\infty $-category. Moreover, the forgetful functor $\operatorname{QCat}^{\asymp } \rightarrow \operatorname{QCat}$ of Notation 8.6.7.1 induces an equivalence of $\infty $-categories $\pi : \operatorname{\mathcal{QC}}^{\asymp } \rightarrow \operatorname{\mathcal{QC}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Proposition 8.6.7.2 to the locally Kan simplicial category $\operatorname{\mathcal{E}}= \operatorname{QCat}$. $\square$