One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which satisfy some commutative diagram relating the Artin maps of $K$ and $L$ when $L$ is a finite extension of $K$ (for example, the existence theorem correspond to the surjectivity of the induced map from the profinite completion).
In the global case, the Artin map becomes a map from the idèle class group $\mathbf{C}_K = \mathbb{A}_K^\times/K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$. My question is the following:
Is it true that, for a global field $K$, the Artin map induces an isomorphism from the profinite completion of $\mathbf{C}_K$ to $\operatorname{Gal}(K^\text{ab}/K)$?
I know that the global Artin map is subjective with kernel $\mathbf{C}_K^\circ$, the identity component of $\mathbf{C}_K$. Hence the question can be reformulated as:
Is it true that the profinite completion of $\mathbf{C}_K$ is equal to $\mathbf{C}_K/\mathbf{C}_K^\circ$?
