Kerodon

Remark 6.3.3.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a reflective localization functor, and let $W$ be the collection of morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $F(w)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Then $W$ is localizing (see Definition 6.2.3.9). To prove this, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is a reflective subcategory of $\operatorname{\mathcal{C}}$ and that $F$ is left adjoint to the inclusion (Remark 6.3.3.11). In this case, $W$ is the collection of $\operatorname{\mathcal{D}}$-local equivalences in $\operatorname{\mathcal{C}}$ (Remark 6.2.2.21), so the result follows from Example 6.2.3.10.