Can you tell me, where I can find a proof of the following fact: Let $R$ be a commutative ring. Consider the category of commutative Hopf algebras over $R$. Then this category is equivalent to the category of corepresentable functors from the category commutative $R$-algebras to groups.
I believe the argument you are looking for may be contained in Jens Carsten Jantzen's book Representations of Algebraic Groups, Part I, Sections 1.3 and 2.3. You may also wish to consult Chapter 1 of William C. Waterhouse's Introduction to Affine Group Schemes.