Let $M=F_{ab}$ the maximal abelian extension of $F$. His Galois group is profinite. If $K$ is a finite abelian subextension, the Artin map gives a surjective morphism $C_F\longrightarrow Gal(K/F)$ from idelic class group of $F$ on Galois group. taking projective limit we construct a morphism from $C_F$ in $Gal(F_{ab}/F)$. It's the definition given by N. Childress in her Class Field Th of the norm residue symbol. I do not see why is it onto and continuous.
If i try to write a bit what happen, on my right i have a compatible collection of automorphismes of abelian sub-extension $(\sigma_K)$. By "compatible" i mean
if $K_1\subset K_2$ then $\sigma_{K_2}\mid_{K_1}=\sigma_{K_1}$
By surjectivity $\forall K, \exists c_K\in C_F,$ such that $\sigma_K={\cal A}_K(c_K)$
where ${\cal A}_K$ is the Artin map defined on idèles $J_F$ or class-group $C_F$.
Compatibility just said if $K\subset L$, ${\cal A}_K(c_K)={\cal A}_K(c_L)$
and i do not see how to obtain a unique class especially since $C_F$ is not compact.