ct.category theory - Derived Functors in arbitrary triangulated categories

Let ${mathcal D}$ be a triangulated category, ${mathcal C}$ a triangulated subcategory and $Q: {mathcal D}to {mathcal D}/{mathcal C}$ the corresponding Verdier-localization. Now suppose we have a triangulated functor ${mathbb F}: {mathcal D}to {mathcal T}$ to some other triangulated category ${mathcal T}$.

My question is the following: Under which circumstances do we have some kind of "right derived" functor of ${mathbb F}$ with respect to ${mathcal C}$? By that I mean a triangulated functor $textbf{R}{mathbb F}: {mathcal D}/{mathcal C}to {mathcal T}$ together with a natural transformation ${mathbb F}Rightarrow textbf{R}{mathbb F}circ Q$ which is initial with this property.

Does there exist such a treatment of derived functors in arbitrary triangulated categories?

Thank you.