Weil group of Lubin-Tate extension

Question

I'm reading this article of local class field theory http://arxiv.org/pdf/math/0606108v2.pdf. I can't understand the proof of prop 4.7(ii)(p. 8), more precisely last sentence. 'It is bijective because it restricts to $ Gal(\widehat K_m^f/\widehat K)=(\mathcal O/\mathfrak p^m)^{*} = \mathcal O^*/(1+p^m)$ by Proposition 4.4(iii) and the quotient $W(\widehat K/K) = Frob^\mathbb Z_K$ is mapped onto $K^*/\mathcal O^*=\mathbb Z$, i.e. $v \circ ρ_{f,m} = v.$' I understood the isomorphism of this Galois group, I can't find out why $W(\widehat K/K)$ maps onto $K^*/\mathcal O^*$ and how it helps us to find Weil's group of $\widehat K_m^f$. I think it should be obviously and I just missed something. I'll be glad for any help.