Corollary 9.2.4.33. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories. Then $F$ is a monomorphism in the $\infty $-category $\operatorname{\mathcal{QC}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be the essential image of $F$. By virtue of Proposition 9.2.4.32, it will suffice to show that $F$ induces an equivalence from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}_0$, which is a reformulation of the requirement that $F$ is fully faithful (Corollary 4.6.2.22). $\square$