Corollary 10.3.5.18. Let $\operatorname{\mathcal{C}}$ be a regular $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory which is closed under finite limits. Assume that, for every morphism $f$ of $\operatorname{\mathcal{C}}_0$, the image $\operatorname{im}(f)$ (formed in the $\infty $-category $\operatorname{\mathcal{C}}$) can be chosen to belong to $\operatorname{\mathcal{C}}_0$. Then $\operatorname{\mathcal{C}}_0$ is also regular.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is regular, $f$ can be factored as the composition of a universal quotient morphism $q: X \twoheadrightarrow Y_0$ with a monomorphism $i: Y_0 \hookrightarrow Y$. If $X$ and $Y$ belong to $\operatorname{\mathcal{C}}_0$, then our assumption guarantees that we can arrange that $Y_0$ is also contained in $\operatorname{\mathcal{C}}_0$. In this case, $i$ is also a monomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_0$, and Proposition 10.3.5.17 guarantees that $q$ is a universal quotient morphism in the subcategory $\operatorname{\mathcal{C}}_0$. Allowing $f$ to vary and invoking Corollary 10.3.5.5, we conclude that $\operatorname{\mathcal{C}}_0$ is regular. $\square$