Corollary 6.2.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, and let $W$ be the collection of $\operatorname{\mathcal{C}}'$-local equivalences in $\operatorname{\mathcal{C}}$. Then an object $C \in \operatorname{\mathcal{C}}$ is $W$-local if and only if it is isomorphic to an object of $\operatorname{\mathcal{C}}'$. In particular, if $\operatorname{\mathcal{C}}'$ is replete, then it coincides with the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Applying Example 6.3.3.11, we see that the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ which exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Proposition 6.2.3.9 implies that the essential image of $\iota $ is the collection of $W$-local objects of $\operatorname{\mathcal{C}}$. $\square$