Remark 6.2.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $u: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. If $u$ is an isomorphism, then it is a $\operatorname{\mathcal{C}}'$-local equivalence. Conversely, if $X$ and $Y$ belong to $\operatorname{\mathcal{C}}'$ and $u$ is a $\operatorname{\mathcal{C}}'$-local equivalence, then it is an isomorphism.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$