# Diagonal functor

In category theory, a branch of mathematics, the **diagonal functor** is given by , which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects *within* the category : a product is a universal arrow from to . The arrow comprises the projection maps.

More generally, given a small index category , one may construct the functor category , the objects of which are called diagrams. For each object in , there is a constant diagram that maps every object in to and every morphism in to . The diagonal functor assigns to each object of the diagram , and to each morphism in the natural transformation in (given for every object of by ). Thus, for example, in the case that is a discrete category with two objects, the diagonal functor is recovered.

Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram , a natural transformation (for some object of ) is called a cone for . These cones and their factorizations correspond precisely to the objects and morphisms of the comma category , and a limit of is a terminal object in , i.e., a universal arrow . Dually, a colimit of is an initial object in the comma category , i.e., a universal arrow .

If every functor from to has a limit (which will be the case if is complete), then the operation of taking limits is itself a functor from to . The limit functor is the right-adjoint of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor.

For example, the diagonal functor described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor. Other well-known examples include the pushout, which is the limit of the span, and the terminal object, which is the limit of the empty category.

## References

- Mac Lane, Saunders; Moerdijk, Ieke (1992).
*Sheaves in geometry and logic a first introduction to topos theory*. New York: Springer-Verlag. pp. 20–23. ISBN 9780387977102.

- May, J. P. (1999).
*A Concise Course in Algebraic Topology*(PDF). University of Chicago Press. p. 16. ISBN 0-226-51183-9.