Diagonal morphism

In category theory, a branch of mathematics, for any object formula_1 in any category formula_2 where the product formula_3 exists, there exists the diagonal morphism 
satisfying 
where formula_7 is the canonical projection morphism to the formula_8-th component. The existence of this morphism is a consequence of the universal property that 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.
For concrete categories, the diagonal morphism can be simply described by its action on elements formula_9 of the object formula_1. Namely, formula_11, the ordered pair formed from formula_9. The reason for the name is that the image of such a diagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphism formula_13 on the real line is given by the line that is the graph of the equation formula_14. The diagonal morphism into the infinite product formula_15 may provide an injection into the space of sequences valued in formula_16; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions that the image of the diagonal map will fail to satisfy.