According to nLab(https://ncatlab.org/nlab/show/monadic+functor), if a functor $U:\mathcal{D}\to\mathcal{C}$ is monadic, the comparison functor induces an equivalence of categories $K^\mathbb{T}:\mathcal{D}\to\mathcal{C}^\mathbb{T}$.
My question is the following: can we describe the quasi-inverse of the comparison functor in an explicit way?
Thank you!
Let us call $F:\mathcal{C}\to \mathcal{D}$ the left adjoint of $U$, so that the endofunctor $T$ of the monad is just $UF$. Then the quasi-inverse $L$ of $K$ can be defined quite simply : its image at an algebra $(X,h)$ in $\mathcal{C}^{\mathbb{T}}$ is the coequalizer of $\epsilon_{FX}$ and $F(h):FUFX\to FX$. Note that this pair of maps is always $U$-split, since its image under $U$ is $\mu_X,T(h):T^2X\to TX$, whose split coequalizer in $\mathcal{C}$ is just $h$. So we can define the functor $L:\mathcal{C}^{\mathbb{T}}\to \mathcal{D}$ as soon as $\mathcal{D}$ has coequalizers for $U$-split pairs, and in fact it is then always the left adjoint of $K$. The other conditions appearing in Beck's theorem, i.e. the preservation and reflection of these coequalizers by $U$ (or preservation of these coequalizers and reflection of isomorphisms), respectively ensure that the unit and counit of the adjunction $L\dashv K$ are isomorphisms.