Remark 8.6.6.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be the opposite fibration. By virtue of Theorem 8.6.5.1 and Corollary 8.6.3.14, $U$ admits a cocartesian dual $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ and a cartesian conjugate $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$, which are uniquely determined up to equivalence and opposite to one another (Proposition 8.6.6.1). When $\operatorname{\mathcal{C}}$ is an $\infty $-category, all four of these fibrations can be realized as a suitable restriction of the projection map $\operatorname{Cospan}(U): \operatorname{Cospan}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. Let $L$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$, and let $R$ denote the collection of all morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then:
Using Proposition 8.1.7.6, we can identify $U$ with the map
\[ \operatorname{Cospan}^{ \mathrm{all}, L \cap R }(\operatorname{\mathcal{E}}) = \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}). \]Using Variant 8.1.7.14, we can identify $U^{\operatorname{op}}$ with the map
\[ \operatorname{Cospan}^{ L \cap R , \mathrm{all}}(\operatorname{\mathcal{E}}) = \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}). \]Using Proposition 8.6.3.5 (and Remark 8.6.3.7), we can identify $U^{\dagger }$ with the projection map $\operatorname{Cospan}^{R,L}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}})$.
Using Corollary 8.6.6.5 (and Remark 8.6.3.7), we can identify $U^{\vee }$ with the map $\operatorname{Cospan}^{L, R}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$.