Kerodon

Remark 6.3.0.5. Let $\operatorname{\mathcal{C}}$ be a category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, 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 $W$. Then, for every category $\operatorname{\mathcal{E}}$, the precomposition functor $\operatorname{Fun}( W^{-1} \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ induces an isomorphism from $\operatorname{Fun}( W^{-1} \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ to the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ spanned by those functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ which carry each element $w \in W$ to an isomorphism in $\operatorname{\mathcal{E}}$. Bijectivity at the level of objects follows immediately from the definition. At the level of morphisms, it follows from the bijectivity of the map