Proposition 6.2.3.8. 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 coincides with the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects.
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Proof. Let $C$ be an object of $\operatorname{\mathcal{C}}$. Assume that $C$ is $W$-local; we wish to show that $C$ is isomorphic to an object of $\operatorname{\mathcal{C}}'$ (the converse follows from Remarks 6.2.3.5 and 6.2.3.4). Since $\operatorname{\mathcal{C}}'$ is reflective, we can choose a $\operatorname{\mathcal{C}}'$-local equivalence $w: C \rightarrow C'$, where $C'$ belongs to $\operatorname{\mathcal{C}}'$. Since $C'$ is also $W$-local, Remark 6.2.3.7 guarantees that $w$ is an isomorphism, so that $C$ is isomorphic to the object $C' \in \operatorname{\mathcal{C}}'$. $\square$