Corollary 10.3.5.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be regular $\infty $-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which preserves finite limits. Then $F$ is regular if and only if, for every morphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$, the map $\operatorname{Sub}(Y) \rightarrow \operatorname{Sub}(F(Y))$ of Remark 10.3.5.11 carries $\operatorname{im}(f)$ to $\operatorname{im}( F(f) )$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$