Remark 9.3.4.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X_0 \rightarrow X$ be a morphism in $\operatorname{\mathcal{C}}$. If $f$ is a monomorphism, then the homotopy class $[f]: X_0 \rightarrow X$ is a monomorphism in the ordinary category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Beware that the converse is false in general.
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