Let $\mathcal L$ a countable language. Let $T$ a coherent $\mathcal L$-theory with no finite models.
Having the following definition of a Back-and-Forth system in mind:
A Back-and-Forth system from $\mathcal A$ to $\mathcal B$ is a non-empty family of partial isomorphisms such that:
(i) for every $\gamma \in \Gamma$ and $a \in A$ there exists an extension $\gamma '$ of $\gamma$ with $a \in dom(\gamma)$
(ii) for every $\gamma \in \Gamma$ and $b \in B$ there exists an extension $\gamma '$ of $\gamma$ with $a \in im(\gamma)$
If any two countable models of $T$, $\mathcal A$ and $\mathcal B$ have a Back-and-Forth system $\Gamma$, how can one prove $T$ is then $\aleph_0$-categorical??
I've thought that, as $\mathcal A$ and $\mathcal B$ have a Bach-and-Forth system, one can get any partial isomorphism and finish constructing/extending a 'full' isomorphism between both universes i.e. $A$ and $B$. And as $\mathcal L, A, B$ are countable maybe the steps taken to construct said isomorphism are countable and somehow proving that $T$ is $\aleph_0$-categorical?
Anyways, thank you very much in advance!
I'm going to assume the corrections noted in my comment above.
Since $\Gamma \neq \emptyset$, choose $\gamma \in \Gamma$ and define $\gamma_0 = \gamma$. Let $a_n, b_n$ enumerate the elements of $A, B$ respectively. Define $\gamma_{n+1}$ inductively as follows:
If $a_{n+1} \notin \operatorname{dom} (\gamma_n)$, choose $(\gamma_n)'$ to be an extension of $\gamma_n$ with $a_{n+1} \in \operatorname{dom}((\gamma_n)')$. Otherwise, define $(\gamma_n)'=\gamma_n$.
If $b_{n+1} \notin \operatorname{image}((\gamma_n)'),$ choose $\gamma_{n+1}$ to be an extension of $(\gamma_n)'$ such that $b_{n+1} \in \operatorname{image}(\gamma_{n+1}).$ Otherwise, define $\gamma_{n+1} = (\gamma_n)'$.
Define $\gamma_\infty = \bigcup \gamma_n$. This definition makes sense because $m \lt n \Rightarrow \gamma_n$ is an extension of $\gamma_m$. By construction, $\operatorname{dom}(\gamma_\infty)=A$ and $\operatorname{image}(\gamma_\infty)=B.$ Then $\gamma_\infty:A \to B$ is an isomorphism. Since, by assumption, any two countable models of $T$ have a back-and-forth system, we've now proved that any two countable models of $T$ have an isomorphism between them, and that means that $T$ is $\aleph_0-$categorical.