Diagonal mapping

In category theory, a branch of mathematics, for any object ${\displaystyle a}$ in any category ${\displaystyle {\mathcal {C}}}$ where the product ${\displaystyle a\times a}$ exists, there exists the diagonal morphism

satisfying

where ${\displaystyle \pi _{k}}$ is the canonical projection morphism to the ${\displaystyle k}$-th component. The existence of this morphism is a consequence of the universal property which characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality.

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