Variant 5.2.8.12 (Enriched Homotopy Transport: Contravariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of $\infty $-categories. Applying Construction 5.2.8.9 to the opposite functor $U^{\operatorname{op}}$, we deduce that there is a unique $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ with the following properties:
For each object $C$ of the $\infty $-category $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (regarded as an object of $\mathrm{h} \mathit{\operatorname{QCat}}$).
For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )^{\simeq } \]is given by the parametrized contravariant transport functor $\operatorname{\mathcal{E}}_{D} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{\mathcal{E}}_{C}$ of Variant 5.2.8.6.
We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the enriched homotopy transport representation of the cartesian fibration $U$. If $U$ is a right fibration, then $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ takes values in the full subcategory $\mathrm{h} \mathit{\operatorname{Kan}} \subseteq \mathrm{h} \mathit{\operatorname{QCat}}$.