Remark 8.6.8.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets. Unwinding the definitions, we see that a diagram $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$ is a contravariant transport representation for $U$ if and only if there exists a commutative diagram of simplicial sets
which exhibits $U$ as a cartesian conjugate of $V$, in the sense of Definition 8.6.1.1. If this condition is satisfied, we say that $T$ exhibits $\mathscr {F}$ as a contravariant transport representation for $U$. In this case, for every vertex $C \in \operatorname{\mathcal{C}}$, we have equivalences of $\infty $-categories $\operatorname{\mathcal{E}}_{C} \xrightarrow {T_ C} \{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \int _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} \leftarrow \mathscr {F}(C)$ (see Example 5.6.2.19).