Localization of a category at a class of morphisms

For a category $\mathcal{C}$ and a class $S$ of its arrows, a localization consists of a category $\mathcal{L}$ and a functor $Q:\mathcal{C}\to \mathcal{L}$ such that $Q(s)$ is invertible for every $s\in S$ and every functor $F:\mathcal{C}\to \mathcal{E}$ inverting $S$ has a unique factorization $F=\bar{F}Q$. We use the strict factorization convention for the same-object roof model. Natural transformations between such functors also descend uniquely; in the equivalence-invariant formulation, precomposition with $Q$ is an equivalence onto the functors inverting $S$. Functors and natural isomorphisms have the meanings of Covariant functor, identity functor, composite functor, and contravariant functor and Natural isomorphism. All category and functor quantifiers use the standing size convention.