So, what is a derived functor?

We rethink the notion of derived functor in terms of corre-spondences, that is, functors ε → [1]. While derived functors in our sense, when they exist, are given by Kan extensions, their existence is a strictly stronger property than the existence of Kan extensions. We show, however, that derived functors exist in the cases one expects them to exist. Our definition is espe-cially convenient for the description of a passage from an adjoint pair (F;G) of functors to a derived adjoint pair (LF;RG). In particular, canonicity of such a passage is immediate in our approach. Our approach makes perfect sense in the context of ∞-categories.