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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?

2.) Is the Hopf algebra functor (also) right adjoint to something?

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    $\begingroup$ 1.) Yes. The functor from {groups} to {Hopf algebras} that sends $G$ to $k\left[G\right]$ is left adjoint to the functor from {Hopf algebras} to {groups} that sends $H$ to the group of all grouplike elements of $H$. $\endgroup$
    – darij grinberg
    Jul 12, 2020 at 10:20

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