Kerodon

Remark 8.6.7.8. Let $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ be the opposition functor of Construction 8.6.7.6. Passing to homotopy categories, we obtain a functor $\overline{\sigma }: \mathrm{h} \mathit{\operatorname{QCat}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$. It follows from Proposition 8.6.7.7 that, up to isomorphism, $\overline{\sigma }$ agrees with the automorphism of $\mathrm{h} \mathit{\operatorname{QCat}}$ which is given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{op}}$, and on morphisms by the construction $[F] \mapsto [F^{\operatorname{op}} ]$; here $[F]$ denotes the isomorphism class of a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $[F^{\operatorname{op}}]$ the isomorphism class of the opposition functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$.