Functors whose domain is a category of morphisms
Connected sequences of functors whose domain, is the category of morphisms of an arbitrary abelian category A and whose range category B is also abelian are compared with the composition functors of Eckmann and Hilton acting between the same categories Sequences of functors of both types are obtained from any half-exact functor A→B if A has enough injectives and projectives.
Pressman, I. (1967). Functors whose domain is a category of morphisms. Acta Mathematica, 118(1), 223–249. doi:10.1007/BF02392482