Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 8.6.7.13. Let $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ denote the $\infty $-category of pairs $(\operatorname{\mathcal{C}}, X)$, where $\operatorname{\mathcal{C}}$ is a small $\infty $-category and $X$ is an object of $\operatorname{\mathcal{C}}$ (see Definition 5.5.6.10). Then the identity functor $\operatorname{id}: \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation for the universal cocartesian fibration

\[ U: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}\quad \quad (\operatorname{\mathcal{C}}, X) \mapsto \operatorname{\mathcal{C}}. \]

Applying Proposition 8.6.7.12, we deduce that the opposition functor $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation for a cocartesian dual of the fibration $U$. By virtue of Corollary 5.6.5.13, this property characterizes the functor $\sigma $ up to isomorphism.