Lemma 8.6.3.6 (04EL)

Lemma 8.6.3.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, let $R$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$, and let $L$ denote the collection of all morphisms $e: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ such that $U(e)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then $\operatorname{Cospan}(U): \operatorname{Cospan}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ restricts to a cartesian fibration of $\infty $-categories $V: \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{C}})$. Moreover, an edge $e: X \rightarrow Y$ of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is $V$-ccartesian if and only if it satisfies the following condition:

$(\ast )$: The edge $e$ corresponds to a cospan $X \xrightarrow {\ell } B \xleftarrow {r} Y$ in $\operatorname{\mathcal{E}}$, where $\ell $ is an isomorphism and $r$ is $U$-cocartesian.

Proof. Let $L_0$ be the collection of all isomorphisms in $\operatorname{\mathcal{C}}$, and let $R_0$ be the collection of all morphisms of $\operatorname{\mathcal{C}}$. Then $L_0$ and $R_0$ are pushout-compatible, in the sense of Definition 8.1.6.5 (Example 8.1.6.6). Moreover, $U$ is a Beck-Chevalley fibration relative to $(R_0, L_0)$ (Example 8.1.10.8). Since $\operatorname{Cospan}^{L_0, R_0}(\operatorname{\mathcal{C}}) = \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}})$ is an $\infty $-category, the desired result follows from Theorem 8.1.10.9 and Remark 8.1.10.11. $\square$