Definition 6.3.0.1. 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 say that $F$ exhibits $\operatorname{\mathcal{D}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if, for every category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$} \} \ar [d] \\ \{ \textnormal{Functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ carrying each $w \in W$ to an isomorphism in $\operatorname{\mathcal{E}}$} \} .} \]