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Remark 5.2.8.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$, and let $F: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ be given by parametrized covariant transport. Then, for every edge $f: C \rightarrow D$, the composite map

$\{ f\} \times \operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \xrightarrow {F} \operatorname{\mathcal{E}}_{D}$

is given by covariant transport along $f$, in the sense of Definition 5.2.2.4. In other words, we can identify $F$ with a diagram $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$, which carries each edge $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ to the covariant transport functor $f_{!}$ of Notation 5.2.2.9.