Given a pair of short exact sequences 1) 0 → X → Y → Z → 0, 0 → A → B → C → 0 in an abelian category A, with sufficiently many projectives and injectives, and given an additive bifunctor T we show that T applied to the pair (1) gives rise to a diagram of a type described by C. T. C. Wall that contains 15 interlocking long exact sequences involving the derived functors of T at (A, X), (A, Y), etc. and also involving the derived functors of Tp and Tq which are two functors with domain A2 that arise through the failure of T to preserve pullbacks and pushouts. In the case of Hom (respectively ø) in the category of G-modules for a group G the derived functors of Tp (respectively Tq) are expressed in terms of group cohomology (respectively homology)
