Kerodon

Remark 8.6.8.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small cartesian fibration of simplicial sets. and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$ be its contravariant transport representation. Then the opposite functor $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is an essentially small cocartesian fibration which admits a covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}^{\operatorname{op}} / \operatorname{\mathcal{C}}^{\operatorname{op}} }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$. In this case, we can identify $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ with the composition $\sigma \circ \operatorname{Tr}_{\operatorname{\mathcal{E}}^{\operatorname{op}}/ \operatorname{\mathcal{C}}^{\operatorname{op}} }$, where $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ is the opposition functor introduced in Construction 8.6.7.6 (given on objects by the construction $\operatorname{\mathcal{D}}\mapsto \operatorname{\mathcal{D}}^{\operatorname{op}}$). This follows by combining Propositions 8.6.6.1 and 8.6.7.12.