Lemma 8.6.3.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then the projection map $V: \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is also a cartesian fibration. Moreover, an edge $X \rightarrow Y$ of $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ is $V$-cartesian if and only if it corresponds to a cospan $X \xrightarrow {\ell } B \xleftarrow {r} Y$ in $\operatorname{\mathcal{E}}$, where $r$ is $U$-cocartesian and $\ell $ is an isomorphism in the $\infty $-category $\{ U(X) \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Proposition 5.1.4.8 and Remark 8.6.3.4, we can reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. In particular, $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, the desired result follows by combining Remark 8.6.3.7 with Lemma 8.6.3.6. $\square$