When trying to understand the construction of the unipotent completion of a finitely generated group $\Gamma$, the following very basic question came to me.
Denote, as usual, $\mathbb{Q}[\Gamma]$ the group ring and $I \subset \mathbb{Q}[\Gamma]$ the augmentation ideal. One looks at $ \varinjlim (\mathbb{Q}[\Gamma]/I^{n+1})^\vee $ but... how are the transition maps defined? What seems natural to me is to project the quotient by $I^n$ into the quotient by $I^{n+1}$ but then, when passing to the dual, one gets maps in the opposite direction. So, how are the maps $ \mathbb{Q}[\Gamma]/I^{n+1} \to \mathbb{Q}[\Gamma]/I^n $ defined?
Thanks for your help!