STATEMENT: Let $K$ be a splitting field for the family $\left\{f_i\right\}_{i\in I}$ and let $E$ be another splitting field. Any embedding of $E$ into $K^a$ inducing the identity on $k$ gives an isomorphism of $E$ onto $K$.
PROOF: Let the notation be as above. Note that $E$ contains a unique splitting field $E_i$ of $f_i$ and $K$ contains a unique splitting field $K_i$ of $f_i$. Any embeddinng $\sigma$ of $E$ into $K^a$ must map $E_i$ onto $K_i$ by $Theorem 3.1$, and hence maps $E$ into $K$.
QUESTION: Why can the author conclude that $E$ maps into $K$ after knowing that the unique splitting fields for each polynomial maps isomorphically into one another. Can't it be the case that it maps into a different field containing $K_i$?