Remark 9.3.4.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$.
If $F$ is fully faithful and $F(f)$ is a monomorphism in $\operatorname{\mathcal{D}}$, then $f$ is a monomorphism in $\operatorname{\mathcal{C}}$.
If $F$ is an equivalence of $\infty $-categories, then $F(f)$ is a monomorphism in $\operatorname{\mathcal{D}}$ if and only if $f$ is a monomorphism in $\operatorname{\mathcal{C}}$.