(mathematics, somewhat informal)Proof in stages, each of which, in general, depends on the previous stages and can be visualized by means of a diagram of morphisms.

2016, Emily Riehl, Category Theory in Context (Aurora: Dover modern math originals)^{[1]}, New York: Dover, →ISBN, OCLC976394474, page x:

Category theory also contributes new proof techniques, such as diagram chasing or arguments by duality; Steenrod called these methods “abstract nonsense.”^{3} The aim of this text is to introduce the language, philosophy, and basic theorems of category theory. A complementary objective is to put this theory into practice: studying functoriality in algebraic topology, naturality in group theory, and universal properties in algebra.

To prove the five lemma, just use diagram chasing.