Composition functors and spectral sequences

then it is fairly well-known that there is an exact sequence for homotopy, homology and cohomology functors which does relate T(g) and T(h) to T( f ) . We regard such an exact sequence both as a special case of the result we aim at and as the axiomatic jumping-off point for the abstract algebraic theory which is developed and applied in this paper. In a previous paper [10], to which this may be regarded as a sequel, we established the machinery of exact couples and spectral sequences in an abelian category 9~. In particular we studied the convergence problem in its fullest generality, and established the exact sequence (Theorem 4.16 of [10]),