# Elementary example of right localization of functor

I am learning about a general framework for derived functors from Hotta et al., D-modules, Perverse Sheaves, and Representation Theory, Appendix B.


A right localization of $$F: \CC \to \DD$$ is a functor $$F_S: \CC_S \to \DD$$ representing $$Fun(\CC_S, \DD) \to Set , \quad G \mapsto Hom_{(\CC,\DD)}(F, G \circ Q).$$

Concretely, this entails the existence of $$\tau: F \to F_S \circ Q$$ such that the composition $$Hom(F_S, G) \to Hom(F_s \circ Q, G \circ Q) \to^{\tau^\ast} Hom(F, G \circ Q)$$ is the identity.

I understand this is used to define right derived functors (when we are localizing with respect to quasi-isomorphisms in the homotopy category) but are there more elementary examples of right (or left) localizations of functors? For instance, if $$\CC$$ is a one-element category enriched over $$\mathrm{Ab}$$ (i.e. a ring) and $$\CC \to \DD$$ is an additive functor, does this notion of localization coincide with classical localization of (noncommutative) rings?