An affine group scheme is defined over a field $k$ to be the representable functor $G: \mathsf{CAlg_k} \to \mathsf{Grp}$. The coordinate ring on $G$, denoted by $\mathcal{O}(G)$, forms a commutative Hopf algebra structure (this can be realized using Yoneda's Lemma). The converse also exists, that given a commutative Hopf algebra $A$, we can find the corresponding affine group scheme, which is the spectra $Spec A(R) = Hom_k(A, R)$. The representation of an affine group scheme is given by a natural transformation of functors $\rho: G \to Aut_V$ where $Aut_V(R) = Aut_R(V \otimes_k R)$ is another functor valued in $\mathsf{Grp}$. If rank $n$ vector space $V$ is free, then $\rho: G \to GL_n$ as $Aut_V \cong GL_n$.  Moreover, the comodules over Hopf Algebra, in this case, we will take $\mathcal{O}(G)$, are defined as usual.

The claim is that there is a canonical equivalence between the category of representations of the affine group scheme $G$ and the category of comodules over the Hopf algebra $\mathcal{O}(G)$

$$ \mathsf{Rep_k(G)} \simeq  \mathsf{Comod_k(\mathcal{O}(G))}.$$ Recall, if there exists a symmetric (tensor), faithful, exact, k-linear fiber functor $\omega: \mathsf{T} \to \mathsf{Vect_k}$, then we call $\mathsf{T}$ a (neutral) Tannakian category $\mathsf{T} \cong \mathsf{Rep_k(G)}$ which is a rigid, k-linear,  tensor category and there exists a fundamental group associated to the Tannakian category which is the affine group $G= Aut^{\otimes}(\omega)$. (One can also realize that the tensor structure on $Rep_k(G)$ is induced by the Hopf algebra.) Overall, the duality between affine group schemes and commutative Hopf algebras is $$ \{\text{affine group schemes over k} \} \leftrightarrow \{ \text{commutative Hopf Algebra over k}\}$$ $$ G \to \mathcal{O}(G)$$ $$ Spec A \leftarrow A$$

Given a fiber functor $\omega: \mathsf{T} \to \mathsf{Vect_k}$, we can get a Hopf algebra $H$ from co-end construction (we will discuss (co)ends shortly on this blog) $$H = \int^{X \in \mathsf{T}} \omega(X)^\vee \otimes_k \omega(X).$$ The matrix coefficients from the coend $H$ can be realized to be associated with the representative functions on $G$ for $\omega$ a forgetful functor. Given a monoidal functor $\omega: \mathsf{T} \to \mathsf{Vect_k}$, one says that $\mathsf{T}$ is equivalent to the category $\mathsf{Comod_k(\mathcal{O}(G))}$ and under this Tannakian hypothesis and ($T \cong \mathsf{Rep_k(G)}$), we have $H \cong \mathcal{O}(G)$. More appropriately, the Hopf algebra structure on $H$ is given by the coend construction, where the (co)multiplication and (co)unit are given by the tensor structure of $\omega$. (I apologize for having rushed this part for this post; please see this, this, and this text for more appropriate definitions and follow-ups.)