Diagonal morphism

In category theory, a branch of mathematics, for any object in any category where the product exists, there exists the diagonal morphism

satisfying

where is the canonical projection morphism to the -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.