Kerodon

Corollary 10.3.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is a universal quotient morphism if and only if $\operatorname{\check{C}}(X/Y)_{\bullet }$ is a universal colimit diagram in $\operatorname{\mathcal{C}}$ (see Definition 7.7.1.15). In particular, if geometric realizations in $\operatorname{\mathcal{C}}$ are universal, then every quotient morphism in $\operatorname{\mathcal{C}}$ is a universal quotient morphism.