Kerodon

Remark 8.6.0.2 (Duality via Transport Representations). Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of small $\infty $-categories (Construction 5.5.4.1). Then $\operatorname{\mathcal{QC}}$ admits an autoequivalence $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$, given on objects by the formula $\sigma (\operatorname{\mathcal{A}}) = \operatorname{\mathcal{A}}^{\operatorname{op}}$ (see Construction 8.6.7.6). Recall that an (essentially small) cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is determined, up to equivalence, by a functor $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, which we refer to as the covariant transport representation of $U$ (Definition 5.6.5.1). In §8.6.7, we show that a cocartesian fibration $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian dual of $U$ if and only if its covariant transport representation is isomorphic to the composition $\operatorname{\mathcal{C}}\xrightarrow { \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} } \operatorname{\mathcal{QC}}\xrightarrow {\sigma } \operatorname{\mathcal{QC}}$ (Proposition 8.6.7.12). This gives another construction of the cocartesian dual of $U$ (albeit one which is cumbersome to work with).