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 relationshipF is left adjoint to G” (or equivalently, “G is right adjoint to F”) is sometimes written

This definition and others are made precise below.

Read more about Adjoint Functors:  Introduction, Formal Definitions, Adjunctions in Full

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Functor Category - Facts
... For instance, if any two objects X and Y in D have a product X×Y, then any two functors F and G in DC have a product F×G, defined by (F×G)(c) = F(c)×G(c) for every object c in C ... and each ηc has a kernel Kc in the category D, then the kernel of η in the functor category DC is the functor K with K(c) = Kc for every object c in C ... have the general rule of thumb that the functor category DC shares most of the "nice" properties of D if D is complete (or cocomplete), then so is DC if D is an abelian category, then so is DC We also ...
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... Every adjunction 〈F, G, ε, η〉 gives rise to an associated monad 〈T, η, μ〉 in the category D ... The functor is given by T = GF ...
Universal Property - Properties - Relation To Adjoint Functors
... to U, then the assignment and defines a functor V from C to D ... φi then define a natural transformation from 1C (the identity functor on C) to UV ... The functors (V, U) are then a pair of adjoint functors, with V left-adjoint to U and U right-adjoint to V ...