Proposition 10.3.5.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory which is closed under the formation of pullbacks, and let $q: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}_0$. If $q$ is a quotient morphism in $\operatorname{\mathcal{C}}$, then it is also a quotient morphism in $\operatorname{\mathcal{C}}_0$. If $q$ is a universal quotient morphism in $\operatorname{\mathcal{C}}$, then it is also a universal quotient morphism in $\operatorname{\mathcal{C}}_0$.
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Proof. Assume that $q$ is a quotient morphism in $\operatorname{\mathcal{C}}$; we will show that it is also a quotient morphism in $\operatorname{\mathcal{C}}_0$ (the analogous assertion for universal quotient morphisms then follows from the criterion of Corollary 10.3.4.7). Since $\operatorname{\mathcal{C}}$ admits pullbacks and $\operatorname{\mathcal{C}}_0$ is stable under the formation of pullbacks, it follows that $\operatorname{\mathcal{C}}_0$ also admits pullbacks. Applying Proposition 10.2.5.6, we deduce that $q$ admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y): \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}_0$, which is also a Čechnerve of $q$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $q$ is a quotient morphism, $\operatorname{\check{C}}_{\bullet }(Z/Y)$ is a colimit diagram in $\operatorname{\mathcal{C}}$ (Proposition 10.3.2.4). It follows that $\operatorname{\check{C}}_{\bullet }(X/Y)$ is also a colimit diagram in $\operatorname{\mathcal{C}}_0$, so that $q$ is a quotient morphism in $\operatorname{\mathcal{C}}_0$. $\square$