Adjoint monads and an isomorphism of the Kleisli categories
Let \({\mathbb{U}}\) be a category, \({\mathbb{F}}\) a monad on \({\mathbb{U}}\) and \({\mathbb{G}}\) a comonad on \({\mathbb{U}}\). If \({\mathbb{F}}\) is right adjoint to \({\mathbb{G}}\), then the Kleisli category \({\mathbb{U}}_{{\mathbb{F}}}\) of \({\mathbb{F}}\) is isomorphic to the Kleisli category \({\mathbb{U}}_{{\mathbb{G}}}\) of \({\mathbb{G}}\), or, alternatively, the category \({\mathbb{U}}_ 0^{{\mathbb{F}}}\) of free F-algebras is equivalent to the category \({\mathbb{U}}_ 0^{{\mathbb{G}}}\) of cofree \({\mathbb{G}}\)-coalgebras. This result is illustrated by an example where \({\mathbb{U}}\) is a category of modules.
scientific article; zbMATH DE number 4152662