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Let $k$ be a field and $\mathcal H$ the category of conilpotent bicommutative Hopf algebras over $k$. (Conilpotent means that every element eventually becomes zero if you apply the reduced comultiplication $\tilde H \to \tilde H \otimes_k \tilde H$ repeatedly, $\tilde H$ being the augmentation ideal of $H$). $\mathcal H$ is an abelian category, and Goerss has constructed a tensor product in "Hopf rings, Dieudonné modules, and $E_*\Omega^2S^3$", Contemp. Math. 239, that represents bilinear maps of such Hopf algebras. He also notes that because of the special adjoint functor theorem, the tensor product has a right adjoint, an internal hom object thus.

What is an explicit description of this internal hom? It seems to me that since $k[\mathbb Z]$ is the unit for the tensor product, and $Hom_{\mathcal H}(k[\mathbb Z],H)$ is the set of grouplike elements of $H$, the internal hom has to be some Hopf algebra whose grouplikes are precisely the Hopf algebra maps.

Edit: $k[\mathbb Z]$ is not actually conilpotent. So the category we should consider (and Goerss in fact considers) is nonnegatively graded Hopf algebras which are group rings in degree $0$.

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