Proposition 8.6.7.11. Let $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ be the opposition functor of Construction 8.6.7.6. Then the restriction $\sigma |_{ \operatorname{\mathcal{S}}}$ is isomorphic to the identity functor from $\operatorname{\mathcal{S}}\subset \operatorname{\mathcal{QC}}$ to itself.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For every $\infty $-category $\operatorname{\mathcal{C}}$, the image $\sigma (\operatorname{\mathcal{C}}) \in \operatorname{\mathcal{QC}}$ is equivalent to the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (Remark 8.6.7.8); in particular, $\operatorname{\mathcal{C}}$ is a Kan complex if and only if $\sigma ( \operatorname{\mathcal{C}})$ is a Kan complex. It follows that $\sigma $ restrict a functor $\sigma _0: \operatorname{\mathcal{S}}\rightarrow \operatorname{\mathcal{S}}$, which is also an equivalence of $\infty $-categories. We wish to show that $\sigma _0$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{S}}}$. This follows from Example 8.4.0.4, since $\sigma _0( \Delta ^0 )$ is homotopy equivalent to the Kan complex $( \Delta ^0)^{\operatorname{op}} \simeq \Delta ^0$. $\square$