Let $H\leq G=\operatorname{Gal}(K/F)$ ($K/F$ is a finite galois extension), why is the following map well defined:
$\varphi:G/H\to\Gamma_F(K^H,K)$ defined by $\sigma H\mapsto\sigma|_{K^H}$ ,where $\Gamma_{F}(K^H,K)$ denotes all homomorphisms from $K^H$ to $K$ that fixes $F$.
My lecture wrote : Let $\sigma\in G$ ,if $\tau\in H$ then $\tau|_{K^H}=\operatorname{Id}_{K^H}$hence $\sigma\tau|_{K^H}=\sigma|_{K^H}$. Why does this imply (what I understand that need to be shown): $\sigma_1|_{K^H}=\sigma_2|_{K^H}\implies\sigma_2^{-1}\sigma_1\in H$ ?
Please note I was not told $H$ is a normal subgroup of $G$.