Definition 8.6.8.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets. We say that a diagram $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$ is a contravariant transport representation for $U$ if $U$ is a cartesian conjugate to the cocartesian fibration $V: \int _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ of Proposition 5.6.2.2.
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