My book (Fraleigh) states the following:
Assumption: All algebraic extensions of $F$ are assumed to be contained in a fixed algebraic closure $\overline{F}$ of $F$.
Isomorphism Extension Theorem- Let $E$ be an algebraic extension of field $F$. Let $\sigma$ be an isomorphism of $F$ onto a field $F'$. Let $\overline{F'}$ be an algebraic closure of $F'$. Then...
As per the assumption, there exists an extension $H$ of field $F$ in which an algebraic closure of $F$ is contained. Similarly, there must be a similar field extension for $F'$, inside which $\overline{F'}$ is contained. What field extension is this. Can we say it is isomorphic to $H'$? How?