What is "five lemma"

n. (context singulare tantum mathematics English) A particular lemma that claims the existence of a particular isomorphism in a commutative diagram given certain other homomorphisms in the diagram.

In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is valid not only for abelian categories but also works in the category of groups, for example.

The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other.