Kerodon

Remark 8.6.7.10 (Uniqueness). Let $\operatorname{Aut}( \operatorname{\mathcal{QC}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{QC}}, \operatorname{\mathcal{QC}})$ spanned by those functors $\operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ which are equivalences of $\infty $-categories. A theorem of Toën ([MR2182378]) guarantees that $\operatorname{Aut}( \operatorname{\mathcal{QC}})$ is a Kan complex having exactly two connected components, each of which is contractible (see Corollary ). Consequently, the opposition functor $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ of Construction 8.6.7.4 is characterized (up to a contractible space of choices) by the fact that it is an equivalence of $\infty $-categories which is not isomorphic to the identity functor $\operatorname{id}_{ \operatorname{\mathcal{QC}}}$.