Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.6.2.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{D}}' \subseteq \operatorname{\mathcal{D}}$ be the essential image of $F$. Then $F$ is fully faithful if and only if it induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$.