 Idempotent monad

A monad is idempotent when its multiplication is invertible. Then $T\eta =\eta T={\mu }^{-1}$. If $(A,a)$ is an algebra for a monad, its unit law and naturality give $a{\eta }_A=1_A$ and ${\eta }_{A}a=Ta{\eta }_{TA}=T(a{\eta }_A)=1_{TA}$. Thus every algebra is isomorphic to a free algebra, and the Kleisli comparison functor into the Eilenberg-Moore category is an equivalence.