Dualizing the statement "A functor is monadic". This is another example of my struggle with the dualizing principle in Category theory. There are two notions, monadicity and comonadicity. I want to see how exactly they are dual to each other. Suppose that the functor $F:\mathcal{C}\to\mathcal{D}$ is monadic. I think I can see that the dual of this statement is NOT that the left adjoint of $F$ (which is assumed to exist by the definition of monadicity) is comonadic - I think I can find an easy counterexample. Is the correct statement that "the opposite functor, $F^{\text{op}}:\mathcal{C}^{\text{op}}\to\mathcal{D}^{\text{op}}$ is comonadic"?

"$G$ is monadic" means $G$ has a left adjoint $F$ such that the comparison functor $c:\mathcal{C}\to (GF)-\mathbf{Alg}$ is an equivalence. Dualizing, $G^{\mathrm{op}}:\mathcal{C}^{\mathrm{op}}\to\mathcal{D}^{\mathrm{op}}$ has a _right_ adjoint $F^{\mathrm{op}}$ such that $c^{\mathrm{op}}$ is an equivalence. $G^{\mathrm{op}}F^{\mathrm{op}}$ is now a _comonad_ on $\mathcal{D}$, since applying $\mathrm{op}$ reverses natural transformations, thus dualizing the monad multiplication $\mu:GFGF\to GF$ and unit $\eta:\mathrm{id}\to GF$ to the structure of a comonad. The codomain of $c^{\mathrm{op}}$ is the Eilenberg-Moore category of coalgebras for this comonad, as one sees directly by comparing the definitions.

Thus the dual of "$G$ is monadic" is "$G^{\mathrm{op}}$ is comonadic."