Kerodon

Remark 11.5.0.92. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets. Then $\mathscr {F}$ is a covariant transport representation of $U$ if and only if there exists a morphism of simplicial sets $G: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ (so that, in particular, the composite map $\operatorname{\mathcal{E}}\xrightarrow {G} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is equal to $U$). In this case, we will say that the morphism $G$ exhibits $\mathscr {F}$ as a covariant transport representation of $U$.