comma category is a representation of Relation to adjoint functors Suppose is a universal morphism from to and is a universal morphism from to . By the universal property of universal morphisms, given any morphism there exists a unique morphism such that the following diagram commutes: If every object of admits a universal morphism to , then the assignment and defines a functor . The maps then define a natural transformation from (the identity functor on ) to . The functors are then a pair of adjoint functors, with left-adjoint to and right-adjoint to . Similar statements apply to the dual situation of terminal morphisms from . If such morphisms exist for every in one obtains a functor which is right-adjoint to (so is left-adjoint to ). Indeed, all pairs of adjoint functors arise from universal constructions in this manner. Let and be a pair of adjoint functors with unit and co-unit (see the article on adjoint functors for the definitions). Then we have a universal morphism for each object in and : For each object in , is a universal morphism from to . That is, for all there exists a unique for which the following diagrams commute. For each object in , is a universal morphism from to . That is, for all there exists a unique for which the following diagrams commute. Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of (equivalently, every object of ). History Universal properties of various topological constructions were presented by Pierre Samuel in 1948. They were later used extensively by Bourbaki. The closely related concept of adjoint functors was introduced independently by Daniel Kan in 1958. See also Free object Natural transformation Adjoint functor Monad (category theory) Variety of algebras Cartesian closed category Notes References Paul Cohn, Universal Algebra (1981), D.Reidel Publishing, Holland. . Borceux, F. Handbook of Categorical Algebra: vol 1 Basic category theory (1994) Cambridge University Press, (Encyclopedia of Mathematics and its Applications) N. Bourbaki, Livre II : Algèbre (1970), Hermann, . Milies, César Polcino; Sehgal, Sudarshan K.. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. Jacobson. Basic Algebra II. Dover. 2009. External links nLab, a wiki project on mathematics, physics and philosophy with emphasis on the