Kerodon

Definition 6.3.0.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We will say that $F$ exhibits $\operatorname{\mathcal{D}}$ as a $1$-categorical localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if, for every category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a fully faithful functor $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, whose essential image consists of those functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ which carry each $w \in W$ to an isomorphism in $\operatorname{\mathcal{E}}$.