I'm trying to read a proof in Dummit and Foote of the statement
Suppose $K/F$ is a Galois extension and $F'/F$ is any extension. Then $KF'/F'$ is a Galois extension, and $Gal(KF'/F') \cong Gal(K/K \cap F')$.
One line I am confused about is
Since $K/F$ is Galois, every embedding of $K$ fixing $F$ is an automorphism of $K$, so the map $\varphi: Gal(KF'/F') \to Gal(K/F$), $\sigma \mapsto \sigma\vert_K$ defined by restricting an automorphism $\sigma$ to the subfield $K$ is well-defined.
I take it this means automorphisms of $KF'$ fixing $F'$ (thus fixing $F$) also send $K$ to $K$? If that's what it means, why is it true?