I have been working through Langlands paper Representations of Abelian Algebraic Groups, and I can't understand why one of his maps is *obvious* and how it helps. First I'll give the notation

Take a algebraic torus over $F$ that splits over a Galois extension $K$ of $F$, then the torus corresponds to a lattice $L$ on which $G=Gal(K/F)$ acts. Then let $\widehat{L}=Hom(L,\mathbb{Z})$. Then we define $\widehat{T}=Hom(\widehat{L},\mathbb{C}^*)$, and we define the Weil group as the extension $$0 \longrightarrow C_K \longrightarrow W_{K/F} \longrightarrow Gal(K/F) \longrightarrow 0.$$ ($C_{K}$ is the idele class group) and finally let $N = \sum_{g \in G} g$

On page 13 of his paper hes shown that there is an isomorphism $$H^{1}(W_{K/F},\widehat{T}) \longrightarrow Hom(H_{1}(W_{K/F},\widehat{L}),\mathbb{C}^{*}).$$ And now he wants to show that that image of a continuous cocycle will be continuous. To do this he proceeds as follows. He defines $U_{K}$ as the elements on norm 1 in $C_K$ and constructs the exact sequence $$1 \longrightarrow U_K \longrightarrow C_K \longrightarrow M_K \longrightarrow 1,$$ $M_K$ being $\mathbb{Z}$ or $\mathbb{R}$. He then uses $L$ to make the sequence $$0 \longrightarrow Hom(L,U_K) \overset{\lambda}\longrightarrow Hom(L,C_K) \overset{\mu}\longrightarrow Hom(L,M_K) \longrightarrow 0$$. And now this is where I get lost, he claims there is an obvious map from $$N(Hom(L,C_K)) \cap Hom(L,U_K)/N(Hom(L,U_K))$$ to $$\hat{H}^{-1}(G,Hom(L,M_K))/ \mu \hat{H}^{-1}(G,Hom(L,C_K)).$$ Here the $\hat{H}$ means Tate groups.

Is this map obvious? and why is the second group finite? which he claims.

Thank you