with image the full subcategory whose objects are the Δ-functors which transform quasi-isomorphisms into isomorphisms. Consequently we can regard a right-(left-)derived functor of F as an initial (terminal) object in the category of Δ-functor morphisms F → G′ (G′ → F ) where G′ ranges over all Δ-functors from J to E which transform quasi-isomorphisms into isomorphisms. As such, the pair (RF, ζ) (or (Lf, ζ)) – if it exists – is unique up to canonical isomorphism.
Let A′ be another abelian category. Any additive functor F : A → A′ extends to a Δ-functor F ̄ : K(A) → K(A′ ). Q′ : K(A′) → D(A′) being the canonical map, we will refer to derived functors of Q′F ̄, or of the restriction of Q′F ̄ to some specified Δ-subcategory J of K(A), as being “derived functors of F” and denote them by RF or LF.