I am struggling with the following general setup from Chapter IV(page 269) in Macdonald's book on symmetric functions and Hall polynomials.
Let $$K=\varprojlim M_n$$ be their inverse limit, which is a profinite group. The character group of $K$ is therefore a discrete group $$\hat{K}=L=\varinjlim\hat{M_n}$$ where $\hat{M_n}$ is the character group of $M_n$. Whenever $m$ divides $n$, $\hat{M_m}$ is embedded in $\hat{M_n}$ by the transpose of the norm homomorphism $N_{n,m}$.
Both groups $L$ and $M$ are (non-canonically) isomorphic to the group of roots of unity in $\mathbb C$ of order prime to $p=\text{char}. k.$
I am thinking of $K=\varprojlim M_n$ as infinite sequences compatible with the norm map $N_{n,m}: M_n\to M_m$ and $\hat{K}=\varinjlim\hat{M_n}$ as elements in the disjoint union of $\hat{M_n}$ modulo the equivalent relation "eventually equal". Then I am having trouble constructing an isomorphism between $\hat K$ and $L$. I suppose that this should be natural, so please also explain the intuition of this isomorphism if possible.
If $G_n$ is a family of finite groups and you have a map $$ \varprojlim G_n \to \mathbb{C}^\times $$ then the map must factor through one of the $G_N$ for some specific $N$ by the no-small-subgroups argument.So to give a map out of $\varprojlim G_n$ is to give a map out of some $G_N$. You could also describe that map as a map out of a $G_{M}$ where $M > N$ in the directed system, and that would count as the same map. This is exactly the definition of the direct limit of the character groups of the $G_i$.