Kerodon

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Remark 6.3.0.9. Let $\operatorname{\mathcal{C}}$ be a category, let $W$ be a collection of morphisms in $\operatorname{\mathcal{C}}$, and let $F: \operatorname{\mathcal{C}}\rightarrow W^{-1} \operatorname{\mathcal{C}}$ be a functor which exhibits $W^{-1} \operatorname{\mathcal{C}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be another functor. Then $G$ exhibits $\operatorname{\mathcal{D}}$ as a $1$-categorical localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if and only if the following conditions are satisfied:

• The functor $G$ carries each $w \in W$ to an isomorphism in $\operatorname{\mathcal{D}}$, and therefore factors uniquely as a composition $\operatorname{\mathcal{C}}\xrightarrow {F} W^{-1} \operatorname{\mathcal{C}}\xrightarrow {G'} \operatorname{\mathcal{D}}$.

• The functor $G': W^{-1} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of categories.