Remark 8.6.3.7 (04EM)

Remark 8.6.3.7. 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 $g$ of $\operatorname{\mathcal{E}}$ such that $U(g)$ is an isomorphism in $\operatorname{\mathcal{C}}$. We then have a commutative diagram of pullback squares

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \ar [d] \ar [r] & \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Cospan}^{\mathrm{all}, R}(\operatorname{\mathcal{E}}) \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \ar [r]^-{\rho _{-}} & \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Cospan}(\operatorname{\mathcal{C}}), } \]

where the vertical map in the middle is a cartesian fibration (Lemma 8.6.3.6), and the horizontal map on the lower left is an equivalence of $\infty $-categories (Proposition 8.1.7.6). It follows that the projection map $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian fibration of $\infty $-categories. Moreover, Corollary 4.5.2.29 implies that the inclusion $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is an equivalence of $\infty $-categories.