# In Theorem VI.1.2 from Lang's *Algebra*, what is the codomain of the map $\sigma \mapsto \sigma|_K$?

I need help understanding what the final sentence below means. Paraphrased from Lang's Algebra, Chapter VI, proof of Theorem 1.2 (page 266, third edition):

Let $$K/k$$ be Galois and $$F$$ an arbitrary extension, $$K,F$$ subfields of some other field. Then $$KF/F$$ and $$K/K\cap F$$ are Galois. Let $$H=\mathrm{Gal}(KF/F)$$, $$G=\mathrm{Gal}(K/k)$$. If $$\sigma \in H$$ then the restriction of $$\sigma$$ to $$K$$ is in $$G$$, and the map $$\sigma \mapsto \sigma |_K$$ gives an injective map of $$H$$ into the Galois group of $$K/K \cap F$$. Let $$H'$$ denote the image of this map.

It's the "and the map..." part I'm not sure about. Reasonably, the map $$\sigma \mapsto \sigma|_K$$ is from $$H$$, but into what? I'm guessing into $$H$$. In the proof it is assumed to be obvious that $$\sigma|_K$$ fixes $$K \cap F$$. If $$H'$$ is the image of $$\sigma \mapsto \sigma|_K$$ ($$H \rightarrow H$$) and $$\sigma \in H'$$, then $$\sigma$$ is supposed to leave $$F$$ unchanged. Since $$K\cap F \subset F$$, that field is also left unchanged. But how does that render the group $$H'$$ to be $$\mathrm{Gal}(K/K\cap F)$$? This group does not leave all of $$F$$ unchanged; nothing is said about $$F\setminus (K\cap F)$$. I don't get this...

Anyway, the result and its proof are not the issue, but what the codomain (target set?) of $$\sigma \mapsto \sigma|_K$$ is, and in particular what can be said about the order of $$\mathrm{Gal}(K/K\cap F)$$ if we know the orders of $$G,H$$.

It's a map into $G=\mathrm{Gal}(K/k)$ - this is hinted at by the statement
If $\sigma \in H$ then the restriction of $\sigma$ to $K$ is in $G$
An element $\sigma\in H=\mathrm{Gal}(KF/F)$ is an isomorphism $$\sigma\colon KF\to KF$$ such that $\sigma(x)=x$ for all $x\in F$. The restriction of $\sigma$ to $K$ will be an isomorphism $\sigma|_K\colon K\to K$, which is not an isomorphism from $KF$ to $KF$, and therefore can't possibly be interpreted as an element of $H$. It is however an element of $\mathrm{Gal}(K/k)$ because $\sigma|_K(x)=x$ for all $x\in k$.
Note that it makes no sense to ask whether or not an element of $\mathrm{Gal}(K/K\cap F)$ "fixes the rest of $F$" or not, because elements of this group are functions that are not defined on parts of $F$ other than $K\cap F$. For example, does the automorphism $\psi:\mathbb{Q}(\sqrt{2})\to\mathbb{Q}(\sqrt{2})$ that sends $\sqrt{2}$ to $-\sqrt{2}$ "fix" $\sqrt{3}$ or not?