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 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?



Elementary example of right localization of functor
Yes, there are many elementary examples, and even in the book you cite, at the start of section B.5, they say "This problem can be easily solved if # = + or − and F is an exact functor" (i.e. if working in the bounded derived category). The theory goes back to a book by Gabriel and Zisman from the 1960s spelling out the language of a Calculus of Fractions, and Quillen's work on model categories from shortly afterwards. This theory is a [categorical version](https://ncatlab.org/nlab/show/calculus+of+fractions) of classical localization in ring theory, and both rely on an Ore condition. 


Classical localization of a ring at a multiplicative subset (or, more precisely, a derived version of localization) is a special case of the calculus of fractions approach. Bill Dwyer nicely spells this out [here](https://www3.nd.edu/~wgd/Dvi/NoncommutativeLocalization.pdf). It also appears in Lurie's Higher Algebra (7.2.4.20), and in [this](https://arxiv.org/pdf/1412.4041.pdf) paper. 


