definition. commuting diagram [kostecki2011introduction, 1.2] [tt-0007]

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A diagram in a category \({\cal C}\) is defined as a collection of objects and arrows that belong to the category \({\cal C}\) with specified operations \(\operatorname {dom}\) and \(\operatorname {cod}\).

A commuting diagram is defined as such diagram that any two arrows of this diagram which have the same domain and codomain are equal.

For example, that the diagram commutes means \(f \mathbin {\bullet } g = h \mathbin {\bullet } k\).

It's also called a arrow diagram [nakahira2023diagrammatic, 1.1] when compared to a string diagram, as it represents arrows with \(\to \).