Proposition 8.6.8.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small cartesian fibration of simplicial sets. Then $U$ admits a contravariant transport representation $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$, which is uniquely determined up to isomorphism. Moreover, the formation of contravariant transport representations induces a bijection
\[ \xymatrix { \textnormal{(Essentially small) cartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} / \textnormal{Equivalence} \ar [d] \\ \{ \textnormal{Diagrams $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$} \} / \textnormal{Isomorphism}. } \]