Proposition 8.6.7.9 (Involutivity). Let $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ be the opposition functor. Then the composition $\sigma \circ \sigma $ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{QC}}}$. In particular, $\sigma $ is an equivalence of $\infty $-categories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. This follows from Proposition 8.6.7.5, since the composition $\widetilde{\sigma } \circ \widetilde{\sigma }$ is equal to the identity functor of the $\infty $-category $\operatorname{\mathcal{QC}}^{\asymp }$. $\square$