Question / exercise goes as follows: $M'$ is said to be finitely generated if there exists a finite $|X|\subset |M'|$ such that $M'$ is the least substructure of $M'$ whose universe $|M'|$ contains $|X|$. We say $M$ is $\omega$-saturated iff every finite $|Y|\subset |M|$, every set of formulas $\Gamma(x)$ from $L(Y)$ (language restricted to that of $Y$) consistent with $\text{Th}(M(Y))$ (i.e. the theory of $M$ restricted to the domain $Y$) is realized in $M(Y)$ (more generally will be realized in the full model $M$).
Show that $M$ is $\omega$-saturated iff given structures $M'$ and $N$ with $M'$ finitely generated and $N$ countable, and given $f:M'\rightarrow M$ and $g:M'\rightarrow N$ elementary embeddings, there exists an elementary embedding $h:N\rightarrow M$ such that $h\circ g = f$.
Now here's my proof of left to right:
Suppose $M$ is $\omega$-saturated, and let $f$, $g$ be defined as above. Now we are given $|N|\le \omega$. Two cases eventuate:
Case 1. $|N|< \omega$. Then given the theory is complete, and two finite models, we have that they are isomorphic, and so that we can set $h = f\circ g^{-1}$.
Case 2. Suppose for the more general case that $|N| = \omega$. Given $|M'| = \omega$ together with $g: M' \rightarrow N$ s.t. $g$ is elementary embedding and $|N| = \omega$, $M' \prec N$ (i.e. $N$ is an elementary extension of $M'$). By our hypothesis that $f: M' \rightarrow M$ is an elementary embedding, we know for any formula $\phi$, $M'\models \phi(m'_1,\dots, m'_n)$, where $m'\in |M'|$ iff $M\models\phi(f(m'_1),\dots,f(m'_n))$, where again $m'\in |M'|$. But $M \prec N$, which means $N \models \phi(f(m'_1),\dots,f(m'_n))$; hence there is an $h: N \rightarrow M$ s.t. $h$ is e.m.
Correct?
Your proof isn't complete. Here are some comments:
In Case 1, you can't literally set $h = f$ ($f$ has domain $M'$, and you want $h$ to have domain $N$!). Rather, you should set $h = f\circ g^{-1}$, which you can do since you know $g$ is an isomorphism.
In Case 2, you say $|M|<\omega$, but you only know $M$ is finitely generated. In fact, there can be no elementary embedding from a finite structure into an infinite one, so we must have $|M| = \omega$.
You need to do more work in Case 2. How do you define $h$?