Eilenberg-Moore comparison functor ($K$)

For $F⊣G:\mathcal{D}\to \mathcal{C}$ with induced monad $T=GF$, the Eilenberg-Moore comparison functor is

When each comparison functor has a left adjoint, iterating the Eilenberg-Moore comparison functor produces the monadic tower. Its monadic length is the least number of comparison steps needed to reach an equivalence; an equivalence has length zero and a non-equivalence that is already monadic has length one.