Remark 8.6.0.1. Ultimately, each of the constructions described above gives rise to essentially the same object. However, it will be useful to consider all three, since each reveals different facets of the overall picture. Fix a cocartesian fibration of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.
The construction of §8.6.2 can be used to show that $U$ admits a cartesian conjugate $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ (Corollary 8.6.2.4). However, from this point of view, it is not obvious that the cartesian fibration $U^{\dagger }$ is unique (unless $\operatorname{\mathcal{C}}$ is assumed to be an $\infty $-category), or that every cartesian fibration can be obtained in this way.
The construction of §8.6.3 produces a cartesian fibration $V: \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ having the property that every cartesian conjugate of $U$ is equivalent to $V$ (Proposition 8.6.3.11). This guarantees that the cartesian conjugate of $U$ is unique up to equivalence. However, it does not by itself prove existence (see Warning 8.6.3.13).
The construction of §8.6.5 can be used to show that $U$ admits a cocartesian dual $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$, which is uniquely determined up to equivalence (Theorem 8.6.5.1). In §8.6.6, we show that the opposite fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ can be realized as a cartesian conjugate of $U^{\vee }$ (Proposition 8.6.6.1). It follows that every cartesian fibration can be realized as the conjugate of a cocartesian fibration. However, from this point of view, it is not obvious that this realization is unique.