Adjoint functors

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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, which helps to minimize the repetition of the same logical details separately in every subject.

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

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.

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