Kerodon

Example 4.4.2.9. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category, and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ be a replete subcategory (Example 4.4.1.12). Then the inclusion map $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is conservative. That is, if $u: X \rightarrow Y$ is a morphism of $\operatorname{\mathcal{C}}$ which is an isomorphism in $\operatorname{\mathcal{D}}$, then $u$ is an isomorphism in $\operatorname{\mathcal{C}}$. To prove this, we observe that if $v: Y \rightarrow X$ is a homotopy inverse of $u$ in the $\infty $-category $\operatorname{\mathcal{D}}$, then the morphism $v$ also belongs to $\operatorname{\mathcal{C}}$ (by virtue of our assumption that $\operatorname{\mathcal{C}}$ is a replete subcategory of $\operatorname{\mathcal{D}}$) and is also a homotopy inverse to $u$ in $\operatorname{\mathcal{C}}$.