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For a right-exact additive functor out of an abelian category with enough projectives, apply the functor to a projective resolution and take homology objects. The zeroth derived functor is $F$, positive derived functors vanish on projective objects, and the degreewise split horseshoe lemma produces a long exact sequence from every short exact sequence in an abelian category. The target can be any abelian category, including an opposite category of one.

Pulling back the final arrow of a short exact sequence in an abelian category preserves its kernel object and yields another short exact sequence in an abelian category. The new kernel inclusion is specified by its old kernel component and zero component into the new quotient. The pullback stability of epimorphisms in an abelian category ensures the new final arrow remains epic.