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

$\newcommand{\CC}{\mathcal C} \newcommand{\DD}{\mathcal D}$ Let $\CC$ be a category and $S$ be a multiplicative set of morphisms satisfying the Ore condition, so that we may form the localization $\CC_S$, with the associated functor $Q: \CC \to \CC_S$. The key definition is:

A

right localizationof $F: \CC \to \DD$ 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?