The idèle \(\alpha = (1,1, \dots, \pi, \dots)\) with a uniformizer \(\pi\) of \(\gotho_{K_\gothp}\) in the \(\gothp\) component and \(1\) elsewhere maps onto the class of \(\gothp\) in \(C_K/U\text{.}\) On the other hand, \(r_K(\alpha) = r_{K, \gothp}(\pi)\) is precisely the Frobenius of \(\gothp\) (because \(L\) is unramified over \(K\)). So indeed \(\gothp\) is being mapped to its Frobenius, and the map \(C_K/U \to \Gal(L/K)\) is indeed Artin reciprocity.