# Kerodon

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Proposition 8.6.7.7. Let $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ be the opposition functor of Construction 8.6.7.6. Then the diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \operatorname{QCat}) \ar [r]^-{\sigma _0} \ar [d] & \operatorname{N}_{\bullet }( \operatorname{QCat}) \ar [d] \\ \operatorname{\mathcal{QC}}\ar [r]^-{\sigma } & \operatorname{\mathcal{QC}}}$

commutes up to isomorphism, where the functor $\sigma _0$ is given by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{op}}$.

Proof. Let $\operatorname{QCat}^{\circ }$ denote the underlying category of the simplicial category $\operatorname{QCat}$, and let us abuse notation by viewing $\operatorname{QCat}^{\circ }$ as a constant simplicial category. We can then identify $\operatorname{QCat}^{\circ }$ with a simplicial subcategory of $\operatorname{QCat}$ having the same objects, with morphism spaces given by

$\operatorname{Hom}_{\operatorname{QCat}^{\circ }}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{sk}_{0}( \operatorname{Hom}_{ \operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } ).$

Note that the inclusion map $\operatorname{QCat}^{\circ } \hookrightarrow \operatorname{QCat}$ factors as a composition

$\operatorname{QCat}^{\circ } \xrightarrow { \iota } \operatorname{QCat}^{\asymp } \xrightarrow { \pi } \operatorname{QCat},$

where the functor $\iota : \operatorname{QCat}^{\circ } \rightarrow \operatorname{QCat}^{\asymp }$ carries each $\infty$-category $\operatorname{\mathcal{C}}$ to itself and each functor of $\infty$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ to the vertex

$\operatorname{id}_{F} \in \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } ) = \operatorname{Hom}_{ \operatorname{QCat}^{\asymp } }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet }.$

We then have a diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \operatorname{QCat}) \ar [r]^-{\sigma _0} \ar [d]^{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\iota )} & \operatorname{N}_{\bullet }( \operatorname{QCat}) \ar [d]^{\operatorname{N}_{\bullet }^{\operatorname{hc}}( \iota ) } \\ \operatorname{\mathcal{QC}}^{\asymp } \ar [d] \ar [r]^-{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \widetilde{\sigma }) } & \operatorname{\mathcal{QC}}^{\asymp } \ar [d] \\ \operatorname{\mathcal{QC}}\ar [r]^-{\sigma } & \operatorname{\mathcal{QC}}, }$

where the upper square is strictly commutative and the lower square commutes up to isomorphism. It follows that the outer rectangle also commutes up to isomorphism. $\square$