Let $\mathsf{Stone}$ denote the category of Stone spaces (compact, totally disconnected Hausdorff spaces) and continuous maps. The forgetful functor $U : \mathsf{Stone} \to \mathsf{Set}$ has a left adjoint $F : \mathsf{Set} \to \mathsf{Stone}$ which maps a set $X$ to the space $F(X)$ of ultrafilters on $X$ (i.e. the Stone-Čech compactification of the discrete space $X$).

**Question.** Is $F$ comonadic?

I have tried to use Beck's monadicity criterion in its dualized form: $F$ is comonadic iff $F$ is a left adjoint ($\checkmark$), $F$ is conservative ($\checkmark$) and $\mathsf{Set}$ has ($\checkmark$) and $F$ preserves equalizers of $F$-split pairs. Only the last point is unclear to me. I hope that it is true, and that actually $F$ preserves much more equalizers.

Here is some background: In his (fantastic!) paper on codensity, Tom Leinster mentions an interesting monad on the category of commutative rings $\mathsf{CRing}$ defined by $T(A) = \prod_{\mathfrak{p} \in \mathrm{Spec}(A)} Q(A/\mathfrak{p})$ and asks for a description for its category of algebras. Each $T(A)$ is von Neumann regular, so I thought it would be good to first look at the restriction to the category of Boolean rings and determine the algebras for the induced monad. By Stone duality, this is monad is dual to the comonad $FU : \mathsf{Stone} \to \mathsf{Stone}$ from above, and comonadicity of $F$ would imply that the boolean $T$-algebras are just products of copies of $\mathbb{F}_2$.