Kerodon

Remark 11.5.0.31. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a reflective localization functor. Then $F$ exhibits $\operatorname{\mathcal{D}}$ as the localization of $\operatorname{\mathcal{C}}$ with respect to some localizing collection of morphisms $W$. The collection $W$ is then uniquely determined: it is the collection of all morphisms $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ for which $F(u)$ is an isomorphism of $\operatorname{\mathcal{D}}$. To prove this, we can assume without loss of generality that $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}[W^{-1}]$ is the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects and that $F$ is a left adjoint to the inclusion functor $\operatorname{\mathcal{C}}[W^{-1}] \hookrightarrow \operatorname{\mathcal{C}}$, in which case it follows from Proposition 9.2.1.18.