Example 8.1.10.13 (04AR)

Example 8.1.10.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are closed under composition and pushout-compatible. Assume that the following conditions are satisfied:

$(0)$: Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{E}}$. If $U(f)$ belongs to $L$, then $f$ is $U$-cartesian. If $U(f)$ belongs to $R$, then $f$ is $U$-cocartesian.
$(1)$: For every object $Y \in \operatorname{\mathcal{E}}$ and every morphism $\overline{f}: \overline{X} \rightarrow U(Y)$ of $\operatorname{\mathcal{C}}$ which belongs to $L$, there exists a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.
$(2)$: For every object $X \in \operatorname{\mathcal{E}}$ and every morphism $\overline{f}: U(X) \rightarrow \overline{Y}$ of $\operatorname{\mathcal{C}}$ which belongs to $R$, there exists a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.

Then $U$ is a dual Beck-Chevalley fibration relative to $(L,R)$. Applying Remark 8.1.10.10, we deduce that the projection map

\[ V: \operatorname{Cospan}(\operatorname{\mathcal{E}}) \times _{\operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) \]

is a locally cartesian fibration. Corollary 8.1.10.12 guarantees that each fiber of $V$ is a Kan complex, so that $V$ is a right fibration (Corollary 5.1.5.12).