Adjunction between groups and complete Hopf algebras

Question

I'm trying to read Quillen's "Rational Homotopy Theory" and at one point he uses that there is an adjunction between the category of groups and complete Hopf algebras, given by $\hat{K}:Grp \to CHA$ and $\mathcal{G}:CHA \to Grp$, where $\hat{K}$ takes a group the completion of the group Hopf algebra $KG$ and $\mathcal{G}$ takes a complete Hopf algebra to its grouplike elements. I'm having trouble understanding this adjunction and would really appreciate some advice. I can see that if we have a map $\hat{K}G \to A$, then the grouplike elements of $\hat{K}G$ must be sent to the grouplike elements of $A$, hence giving us a map $G \to \mathcal{G}A$. However, I am having trouble with the other direction.