The kernel of the reciprocity map in global class field theory

I am learning Class Field Theory by reading Milne's notes and Neukirch's book. There is a proof I can't find.

Let $K$ be a number field. One constructs the map $I_K \rightarrow G_K^\text{ab}$ using local class field theory then shows that it factors through $C_K/C_K^0 = I_K/ \overline{(K^\times.K_\infty^\times)^\circ} \to G_K^\text{ab}$. My question is:

How to prove that this map is injective, surjective and that it is an homeomorphism?

In Milne's note (remark 5.7, p. 174), he claims that is is bijective without proving it (or am I missing something?).

The global reciprocity map kills the discrete copy of $K^\times$, so we can think of it as a continuous homomorphism $\psi:C_K\rightarrow\mathrm{Gal}(K^{ab}/K)$.
In the number field case, $\psi$ is surjective (using compactness of the norm-$1$ idele class group and the continuous idelic norm map) but not injective. Its kernel is the identity component of $C_K$, which is the intersection of all open subgroups of finite index.
So, $\psi$ is not bijective. What is bijective is the induced continuous map from the profinite completion of $C_K$ and $\mathrm{Gal}(K^{ab}/K)$ (which is a homeomorphism because it is bijective and continuous with compact source and Hausdorff target). The point is that the map from $C_K$ to its profinite completion, which, by definition, has kernel equal to the intersection of all open subgroups of finite index, is not injective, i.e., the aforementioned intersection is not trivial.
For every finite abelian extension $L$ of $K$, the Artin map defines an isomorphism $C_{K}/Nm(C_{L})\rightarrow \mathrm{Gal}(L/K)$. When we pass to the inverse limit over $L$, we get an isomorphism with $\mathrm{Gal}(K^{\mathrm{ab}}/K)$ on the right, so the problem is to compute the inverse limit of the system $C_{K}/Nm(C_{L})$. The existence theorem shows that the groups $Nm(C_{L})$ are exactly the open subgroups of finite index, so this is a problem in topology.