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Adjoint functors
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===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.
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