Corollary 8.6.6.3. For every simplicial set $\operatorname{\mathcal{C}}$, the formation of cartesian conjugates induces a bijection
\[ \xymatrix@C =50pt@R=50pt{ \{ \textnormal{Cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} / \textnormal{Equivalence} \ar [d] \\ \{ \textnormal{Cartesian fibrations $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$} \} / \textnormal{Equivalence.} } \]