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Construction 5.2.5.2 (The Homotopy Transport Representation: Covariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote the homotopy category of $\infty$-categories. It follows from Proposition 5.2.5.1 and Example 5.2.2.5 that there is a unique functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ with the following properties:

• 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 $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( [f] ) = [ f_{!} ]$. Here $[f_{!}]$ denotes the isomorphism class of the covariant transport functor of Notation 5.2.2.9, which we regarded as an element of the set

$\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} ) = \pi _0( \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D})^{\simeq } ).$

We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the homotopy transport representation of the cocartesian fibration $U$.