Corollary 9.1.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a left adjoint to the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$. Let $W$ be the collection of all morphisms $w: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ for which $L(w)$ is an isomorphism in $\operatorname{\mathcal{C}}'$. Then:
- $(1)$
The collection $W$ is localizing (Definition 9.1.1.15).
- $(2)$
Every object of $\operatorname{\mathcal{C}}'$ is $W$-local (Definition 9.1.1.1).
- $(3)$
If $\operatorname{\mathcal{C}}'$ is replete, then every $W$-local object of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}'$.