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.
The proofs of these statements can be found, e.g., in the Artin-Tate notes on class field theory. Chapter 9 is about the identity component, but the fact that it is the kernel of the reciprocity map might actually be proved somewhere in the later chapter about abstract class formations. I don't have access to the book right now so I can't say exactly where it is, but I know it's in there.