Inverse limit of Hopfalgebras My Question relates to Corollary 2.7 of
http://www.jmilne.org/math/xnotes/tc.pdf
So Let $k$ be a field and $\mathbb{G}_i$ be an projective system of affine $k$-groupschemes.
I want to know if the inverse limit in the category of affine $k$-groupschemes is the same as the inverse limit in the category of the affine k-schemes, i.e. if the corresponding  inductive limit of the $k$-Hopfalgebras (Deligne calls them Bialgebras in the link above) is the inductive limit of $k$-Algebras.
More precisely i want to know if there is an isomorphism (of groups or of sets) between the inverse limit of $k$-rational points and the $k$-rational points of the inverse limit, i.e.
$(\underset{\leftarrow}{\lim}\mathbb{G}_i)(k)\cong\underset{\leftarrow}{\lim}(\mathbb{G}_i(k))$.
And if that is not the case, is there at least a canonical morphism between those two.
 A: From the definition of limit we have $\text{Hom}(A,\lim B_i) \cong \lim \text{Hom}(A,B_i)$. This holds in any category. In particular, your final equation is certainly true (as an iso of sets). 
Moreover, I feel like an argument of the form "limits commute with limits" ought to show that it is an iso of groups as well (because the definition of group object involves some limits), but I'm doing math on an empty stomach, so I'm not too sure. I apologize.
EDIT:
It's abstract nonsense. Given a complete category $C$, and denoting the associated cat of groups $D$, and the forgetful functor by $F{:}C \rightarrow D$, you can take the limit $\lim F G_i$ and endow it with a group structure in such a way that the maps $\lim FG_i \rightarrow FG_i$ become group homomorphisms, from which it follows that $F$ preserves limits.
For instance, let $f,g{:}G \rightarrow H$ be group homomorphisms and $E \rightarrow G$ their equalizer in $C$. Then the obvious morphism $E \times E \rightarrow G \times G \rightarrow G$ equalizes $f$ and $g$ (use that $f$ and $g$ are group homomorphisms), so we obtain a map $E \times E \rightarrow E$. Similarly one obtains an inverse and a unit, and the group axioms are easy to verify. Moreover, $E \rightarrow G$ is a group homomorphism, so it follows that $E \rightarrow G$ is an equalizer for $f$ and $g$ in $D$. 
One goes through the same stuff for (infinite) products, and hence obtains that $F{:}C \rightarrow D$ preserves limits. Apply it to the category of affine $k$-schemes, and you find an affirmative answer to your question.
