There is a proof given here (Theorem 1.3.4) of the equivalence between the categories $\mathbf{AffGrpSch}/k$ and $\mathbf{HopfAlg}^{opp}_k$.
The author writes in the line before the theorem that there is a canonical functor $\mathbf{AffGrpSch}/k \to \mathrm{Fun}(k-\mathbf{Alg}, \mathbf{Set})$. Is this a mistake? Because in the diagram in the statement of the theorem the functor is $\mathbf{AffGrpSch}/k \to \mathrm{Fun}(k-\mathbf{Alg}, \mathbf{Grp})$.
How is the functor $\mathrm{Fun}($k$-\mathbf{Alg}, \mathbf{Set}) \to \mathrm{Fun}($k$-\mathbf{Alg}, \mathbf{Grp})$ defined in the diagram? Specifically, what does 'induced by the forgetful functor' mean in the right side of the diagram?
Another question, on page 9 would it not make sense to write $\mathbf{HopfAlg} \to \mathrm{Fun}($k$-\mathbf{Alg}, \mathbf{Grp})$ after the author writes that we write $h^A$ for the functor $k-\mathbf{Alg} \to \mathbf{Grp}$?