Kleisli Category - Kleisli Adjunction

Kleisli Adjunction

Kleisli categories were originally defined in order to show that every monad arises from an adjunction. That construction is as follows.

Let〈T, η, μ〉be a monad over a category C and let C_T be the associated Kleisli category. Define a functor F : C → C_T by

and a functor G : C_T → C by

One can show that F and G are indeed functors and that F is left adjoint to G. The counit of the adjunction is given by

Finally, one can show that T = GF and μ = GεF so that 〈T, η, μ〉is the monad associated to the adjunction 〈F, G, η, ε〉.