Proposition 10.3.5.16. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which admit finite limits and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which preserves finite limits and geometric realizations of simplicial objects. Then $F$ carries quotient morphisms of $\operatorname{\mathcal{C}}$ to quotient morphisms of $\operatorname{\mathcal{D}}$. In particular, if the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are regular, then the functor $F$ is regular.
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Proof. Let $f: X \twoheadrightarrow Y$ be a quotient morphism in $\operatorname{\mathcal{C}}$; we wish to that $F(f)$ is quotient morphism in $\operatorname{\mathcal{D}}$. Let $\operatorname{\check{C}}_{\bullet }(X/Y): \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ be a Čechnerve of $f$ (Notation 10.2.5.5. Since $f$ is a quotient morphism (Remark 10.3.4.2), $\operatorname{\check{C}}_{\bullet }(X/Y)$ is a colimit diagram in $\operatorname{\mathcal{C}}$ (Proposition 10.3.2.4). Our assumption that $F$ preserves geometric realizations guarantees that $F \circ \operatorname{\check{C}}_{\bullet }(X/Y)$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. Since $F$ preserves finite limits, $F \circ \operatorname{\check{C}}_{\bullet }(X/Y)$ is a Čechnerve of the morphism $F(f): F(X) \rightarrow F(Y)$. Applying Proposition 10.3.2.4 again, we deduce that $F(f)$ is a quotient morphism in $\operatorname{\mathcal{D}}$. $\square$