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On mathoverflow the following question appeared and someone here might know the answer:
Suppose that $G$ is a group and $k$ is a field. Then it is well known that the group ring (group algebra) functor $k[\bullet]$ is left adjoint to the group of units functor, the latter of which associates the group of units to each ring. This implies that every group morphism from $G$ into the group of units of an algebra $A$ can be uniquely extended into an algebra morphism from $k[G]$ to $A$.
Now, since $k$ is a field, $k[\bullet]$ can also be seen as the Hopf group algebra. My question is now twofold:
1.) Is the Hopf algebra functor still left adjoint to something? Hence inherits similar extension properties? (Answered with yes already)
2.) Is the Hopf algebra functor (also) right adjoint to something? <= This is the actually interesting part.
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