Theorem 8.1.10.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pushouts, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a dual Beck-Chevalley fibration and let $R$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. Then the map $\operatorname{Cospan}(U): \operatorname{Cospan}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}( \operatorname{\mathcal{C}})$ restricts to a cartesian fibration of $\infty $-categories
Moreover, a morphism of $\operatorname{Pith}( \operatorname{Cospan}^{ \mathrm{all}, R}( \operatorname{\mathcal{E}}) )$ is $V$-cartesian if and only if it corresponds to a cospan $X \xrightarrow {f} B \xleftarrow {g} Y$ in the $\infty $-category $\operatorname{\mathcal{E}}$, where $f$ is $U$-cartesian and $g$ is $U$-cocartesian.