Corollary 8.1.10.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are closed under composition and pushout-compatible. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a dual Beck-Chevalley fibration with respect to $(L,R)$, let $\widetilde{L}$ denote the collection of all $U$-cartesian morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f) \in L$, and let $\widetilde{R}$ denote the collection of all $U$-cocartesian morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f) \in R$. Then:
- $(1)$
The simplicial set $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is an $(\infty ,2)$-category.
- $(2)$
The morphism $U$ induces a right fibration $\operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$.
- $(3)$
If $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is an $\infty $-category, then $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is an $\infty $-category, and $U$ induces a right fibration $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$.