Kerodon

Remark 8.1.10.11. In the situation of Theorem 8.1.10.9, suppose that $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is an $\infty $-category (this condition is satisfied, for example, if either $L$ or $R$ consists of isomorphisms; see Proposition 8.1.7.5). Then every $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is thin. Applying Remark 8.1.9.11, we deduce that every $2$-simplex of $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is thin, so that $\operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }(\operatorname{\mathcal{E}})$ is also an $\infty $-category (Example 2.3.2.4). In this case, Theorem 8.1.10.9 asserts that the projection map $V: \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{E}})$ is a cartesian fibration.