Corollary 8.1.10.5. 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, let $L$ be the collection of all $U$-cartesian morphisms of $\operatorname{\mathcal{C}}$, and let $R$ be the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{C}}$. Then $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is an $(\infty ,2)$-category, and $U$ induces a right fibration of $\infty $-categories $\operatorname{Pith}(\operatorname{Cospan}^{L,R}( \operatorname{\mathcal{E}})) \rightarrow \operatorname{Pith}(\operatorname{Cospan}(\operatorname{\mathcal{C}}))$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The collections $L$ and $R$ are closed under composition (Corollary 5.1.2.4) and pushout-compatible by virtue of our assumption that $U$ is a Beck-Chevalley fibration. Using Proposition 8.1.6.7, we see that $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is an $(\infty ,2)$-category. Moreover, the pith of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ can be identified with the subcategory of $\operatorname{Pith}( \operatorname{Cospan}^{ \mathrm{all}, R}(\operatorname{\mathcal{E}}) )$ spanned by those morphisms which are cartesian with respect to the fibration $V: \operatorname{Pith}( \operatorname{Cospan}^{ \mathrm{all}, R}( \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Pith}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) )$ of Theorem 8.1.10.3. The desired result now follows from Corollary 5.1.4.16. $\square$