The five lemma is one of the basic lemmas of homological algebra, useful for example in the construction of the connecting homomorphism in the homology long exact sequence. Let be an abelian category. Consider a commutative diagram in of the form where the top and bottom rows are exact sequences. For simplicity we denote all the differentials in both exact sequences by . on terminology The weak four lemma is another terminology (cf. MacLane, Homology) for the same as 1.1 and 1.2 except that in..