Kerodon

Example 8.6.3.3. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category, so the projection map $\operatorname{\mathcal{E}}\rightarrow \Delta ^{0}$ is a cocartesian fibration of simplicial sets. Then the simplicial set $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\Delta ^0)$ of Notation 8.6.3.1 can be identified with the simplicial set $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}})$ of Construction 8.1.7.2. In particular, $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\Delta ^0)$ is an $\infty $-category (Proposition 8.1.7.5), and Proposition 8.1.7.6 supplies an equivalence of $\infty $-categories $\rho _{+}: \operatorname{\mathcal{E}}\rightarrow \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\Delta ^0)$.