Editing Adjoint functors (section)

==Relationships==

===Universal constructions===

As stated earlier, an adjunction between categories ''C'' and ''D'' gives rise to a family of [[universal morphism]]s, one for each object in ''C'' and one for each object in ''D''. Conversely, if there exists a universal morphism to a functor ''G'' : ''C'' → ''D'' from every object of ''D'', then ''G'' has a left adjoint.

However, universal constructions are more general than adjoint functors: 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 ''D'' (equivalently, every object of ''C'').

===Equivalences of categories===

If a functor ''F'' : ''D'' → ''C'' is one half of an [[equivalence of categories]] then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.

Every adjunction 〈''F'', ''G'', ε, η〉 extends an equivalence of certain subcategories. Define ''C''<sub>1</sub> as the full subcategory of ''C'' consisting of those objects ''X'' of ''C'' for which ε<sub>''X''</sub> is an isomorphism, and define ''D''<sub>1</sub> as the [[full subcategory]] of ''D'' consisting of those objects ''Y'' of ''D'' for which η<sub>''Y''</sub> is an isomorphism. Then ''F'' and ''G'' can be restricted to ''D''<sub>1</sub> and ''C''<sub>1</sub> and yield inverse equivalences of these subcategories.

In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse of ''F'' (i.e. a functor ''G'' such that ''FG'' is naturally isomorphic to 1<sub>''D''</sub>) need not be a right (or left) adjoint of ''F''. Adjoints generalize ''two-sided'' inverses.

===Monads===

Every adjunction 〈''F'', ''G'', ε, η〉 gives rise to an associated [[monad (category theory)|monad]] 〈''T'', η, μ〉 in the category ''D''. The functor
:<math>T : \mathcal{D} \to \mathcal{D}</math>
is given by ''T'' = ''GF''. The unit of the monad
:<math>\eta : 1_{\mathcal{D}} \to T</math>
is just the unit η of the adjunction and the multiplication transformation
:<math>\mu : T^2 \to T\,</math>
is given by μ = ''G''ε''F''. Dually, the triple 〈''FG'', ε, ''F''η''G''〉 defines a [[comonad]] in ''C''.

Every monad arises from some adjunction—in fact, typically from many adjunctions—in the above fashion. Two constructions, called the category of [[Eilenberg–Moore algebra]]s and the [[Kleisli category]] are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad.