Let $K$ be a number field, $L/K$ finite Galois, $\Bbb I_K$ and $\Bbb I_L$ their respective idèles and $C_K, C_L$ their idèle class groups. Let $N_{L/K} : \Bbb I_L \to \Bbb I_K$ be the norm on the idèles induced componentwise by the local norms coming from the $L_w/K_v$ for $w \mid v$ for all places $v$ of $K$ and $N_{L/K} : C_L \to C_K$ the induced norm on the idèle class group of $L$.
I'm struggling to see why $\Bbb I_K/K^\times \cdot N_{L/K} \Bbb I_L \cong C_K/N_{L/K}C_L$. The source I'm reading suggests that one could use the Snake lemma on the diagram coming from the exact sequences
$$0 \to L^\times \to \Bbb I_L \to C_L \to 0$$
and
$$0 \to K^\times \to \Bbb I_K \to C_K \to 0$$
with the vertical maps coming from the norm maps described above. Perhaps a silly moment, but I can't see how this gives me the desired isomorphism.