Construction 5.2.5.7 (The Homotopy Transport Representation: Contravariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets and let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote the homotopy category of $\infty $-categories (Construction 4.5.1.1). It follows from Proposition 5.2.5.1 and Example 5.2.2.5 that there is a unique functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ satisfying the following conditions:
For each vertex $C$ of the simplicial set $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ 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 each edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$ representing a morphism $[f] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(C,D)$, we have $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( [f] ) = [ f^{\ast } ]$, where $[f^{\ast }]$ denotes the isomorphism class of the contravariant transport functor of Notation 5.2.2.18.
We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the homotopy transport representation of the cartesian fibration $U$.