The Barr-Beck monadicity theorem gives necessary and sufficient conditions for a category $\mathcal{C}$ to be equivalent to a category of (co)algebras over a (co)monad. A functor $F:\mathcal{C}\to\mathcal{D}$ is said to be **comonadic** if it has a right adjoint $G$ and the "enhanced functor" $F': \mathcal{C}\to\mathsf{Coalg}_{FG} (\mathcal{D})$ into colagebras over the comonad $FG$ on $\mathcal{D}$ is an equivalence. The comonadic version of Barr-Beck says:

A functor $F:\mathcal{C}\to\mathcal{D}$ is comonadic if and only if:

- F has a right adjoint;
- F reflects isomorphisms;
- Every $F$-cosplit pair in $\mathcal{C}$ admits an equaliser in $\mathcal{C}$ and it is preserved by $F$.

I have recently learned that this substantially simplifies the proof of Tannaka duality for a (neutral) Tannakian category $\mathcal{C}$ with fiber functor $\omega:\mathcal{C}\to\mathsf{Vecf}_k$ to be equivalent to the category of finite-dimensional representations of an affine group scheme $G$. Namely, the Barr-Beck theorem applied to $\omega$ establishes that $\mathcal{C}$ is equivalent to the category of comodules over a coalgebra $A$. Then the rigid tensor structure on $\mathcal{C}$ can be used to show that $A$ is a commutative Hopf algebra and hence is the coordinate ring of an affine group scheme.

I would like some tips towards the Barr-Beck part of this proof.

Firstly, note that by definition $\omega$ is exact and faithful, so conditions (2) and (3) in Barr-Beck are satisfied. I am not sure how to easily show condition (1), that $\omega$ has a right adjoint $R$ (adjoint functor theorems seem too general, but maybe I have missed something). Can anyone help with this?

Secondly, assume that Barr-Beck applies and that $\omega'$ induces an equivalence $\mathcal{C}\simeq\mathsf{Coalg}_{C} (\mathsf{Vecf}_k)$, where $C = \omega\circ R$ is the comonad induced on $\mathsf{Vecf}_k$ by the adjunction $\omega\dashv R$. Then $A=C(k)$ is a $k$-coalgebra, and I believe that $\mathsf{Coalg}_C (\mathsf{Vecf}_k)$ is equivalent to the category $\mathsf{Comodf}_A$ of finitely-generated comodules over the coalgebra $A$. However, I am not sure how to do this because I don't know whether $k$ can be made into a $C$-coalgebra $a_k: k\to C(k)$. If it can, I would want to define the $A$-comodule structure on a $C$-coalgebra $V$ as

$$V\xrightarrow{\sim} k\otimes V\xrightarrow{a_k\otimes \text{id}} A\otimes V,$$

but I don't know if this is correct. Can anyone confirm or correct me on this, and possibly sketch out the details of the equivalence

$$\mathsf{Coalg}_C (\mathsf{Vecf}_k) \simeq \mathsf{Comodf}_A$$

that would complete this part of the proof? Thanks for your help!