Question about a step in the proof of Theorem 8.12 in Morandi's Field and Galois Theory

In the proof of Theorem 8.12 in Morandi's Field and Galois Theory, the author writes:

Because $KN$ is the composite of a Galois extension of $S$ with a purely inseparable (hence normal) extension, $KN/S$ is normal. Thus, $\sigma_j(K)\subseteq KN$ by Proposition 3.28.

I do not understand this implication.

At this point, $\sigma_j$ is only an $F$-homomorphism. Proposition 3.28 is a statement about $E$-homomorphisms when $L/E$ is a normal extension. Since $\sigma_j$ is not assumed to be an $S$-homomorphism, it seems that the hypotheses of Proposition 3.28 are not satisfied.

Has anyone worked through this proof or read this part of the book? If so, could you explain why Proposition 3.28 can be applied here?

Alternatively, is the following statement true?

Let $K/F$ be a finite extension, let $S$ be the separable closure of $F$ in $K$, and let $N$ be the normal closure of $S/F$. Is it true that $KN/F$ is a normal extension?

If so, then the conclusion $\sigma_j(K)\subseteq KN$ would seem to follow immediately from Proposition 3.28.

After looking through other questions on Math Stack Exchange, I found reuns's answer to the following question:

Is $F\hat{E_s}/E$ normal, where $F/E$ is a finite field extension?

According to his answer, $KN/F$ is not a normal extension in general. Therefore, this does not seem to justify the application of Proposition 3.28.