Construction 8.6.7.6 (The Opposite Functor). The simplicial category $\operatorname{QCat}^{\asymp }$ is equipped with an automorphism $\widetilde{\sigma }$, given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{op}}$ and on morphism spaces by the composition
\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{QCat}^{\asymp } }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } & = & \operatorname{Tw}( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } ) \\ & \simeq & \operatorname{Tw}( (\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } )^{\operatorname{op}} ) \\ & \simeq & \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}})^{\simeq } ) \\ & = & \operatorname{Hom}_{ \operatorname{QCat}^{\asymp } }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} ), \end{eqnarray*}
where the isomorphism on the second line is supplied by Remark 8.1.1.7. It follows from Proposition 8.6.7.5 that there exists a functor of $\infty $-categories $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ for which the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{QC}}^{\asymp } \ar [d]^{\pi } \ar [r]^-{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \widetilde{\sigma } ) }_{\sim } & \operatorname{\mathcal{QC}}^{\asymp } \ar [d]^{\pi } \\ \operatorname{\mathcal{QC}}\ar [r]^-{\sigma } & \operatorname{\mathcal{QC}}} \]
commutes up to isomorphism. Moreover, the functor $\sigma $ is uniquely deterrmined up to isomorphism. We will refer to $\sigma $ as the opposition functor for the $\infty $-category $\operatorname{\mathcal{QC}}$.