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Proposition 8.6.6.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. Then $U^{\dagger }$ is a cartesian conjugate of $U$ (in the sense of Definition 8.6.1.1) if and only if the opposite fibration $U^{\dagger , \operatorname{op}}: \operatorname{\mathcal{E}}^{\dagger ,\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian dual of $U$ (in the sense of Definition 8.6.3.1).

Proof of Proposition 8.6.6.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration. Let $U^{\vee }: \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection map. Then $U^{\vee }$ is a cocartesian dual of $U$ (Theorem 8.6.5.6), and the opposite fibration $U^{\vee , \operatorname{op}}$ is a cartesian conjugate of $U$ (Corollary 8.6.6.7). Let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration of simplicial sets. Using the uniqueness assertions of Theorem 8.6.4.1 and Corollary 8.6.6.8, we see that the following conditions are equivalent:

  • The fibration $U^{\dagger }$ is a cartesian conjugate of $U$.

  • The fibration $U^{\dagger }$ is equivalent to $U^{\vee , \operatorname{op}}$ (as a cartesian fibration over $\operatorname{\mathcal{C}}^{\operatorname{op}}$).

  • The fibration $U^{\dagger , \operatorname{op}}$ is equivalent to $U^{\vee }$ (as a cocartesian fibration over $\operatorname{\mathcal{C}}$).

  • The fibration $U^{\dagger , \operatorname{op}}$ is a cocartesian dual of $U$.

$\square$