Kerodon

Corollary 8.1.6.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. Let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are pushout-compatible, contain all isomorphisms of $\operatorname{\mathcal{C}}$, and are closed under composition. Then $X$ and $Y$ are isomorphic as objects of the $\infty $-category $\operatorname{\mathcal{C}}$ if and only if they are isomorphic as objects of the $(\infty ,2)$-category $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$.