Adjoint Functors

In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency. It is studied in generality by the branch of mathematics known as category theory.

In the most concise symmetric definition, an adjunction between categories C and D is a pair of functors,

and

and a family of bijections

which is natural in the variables X and Y. The functor F is called a left adjoint functor, while G is called a right adjoint functor. The relationship “F is left adjoint to G” (or equivalently, “G is right adjoint to F”) is sometimes written

This definition and others are made precise below.