Kerodon

Remark 8.1.9.2. In the situation of Proposition 8.1.9.1, let $C$ be a vertex of $\operatorname{\mathcal{C}}$ and let $\operatorname{\mathcal{E}}_{C}$ denote the fiber $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Note that a morphism $u$ in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ is an isomorphism if and only if it is $U$-cocartesian when viewed as a morphism in $\operatorname{\mathcal{E}}$ (Proposition 5.1.4.12). It follows that the fiber $\{ C\} \times _{ \operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{\mathrm{all},W}(\operatorname{\mathcal{E}})$ can be identified with the $\infty $-category $\operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}_{C} )$. In particular, Proposition 8.1.7.6 supplies an equivalence of $\infty $-categories