Let $T$ be a countably categorical first order theory. Let $M \vDash T$ be arbitrary and countable.
Show that there exists $\{M_n : n \in \omega \}$ of countable models of $T$ such that $M_{n+1} \prec M_n$ ($M_{n+1}$ is a proper elementary substructure of $M_n$), and $M = \cap_{n < \omega}M_n$.
I don't think it's as simple as doing a bunch of Downwards Lowenheim Skolem. I was thinking of defining a model $B = (N, P_i)_{i \in \omega}$ where each predicate will eventually be interpreted as $M_i$ and say that, for every formula $\phi$, $B$ is a model of $P_{i+1}(a) \implies (\phi(a)^{P_{i+1}} \iff \phi(a)^{P_i})$ where the superscript indicates the relativization of the formula. $B$ will also model other sentences indicating $P_{i+1} \subset P_i$ and the $P_i$ are models, etc. But I'm not sure how to fill the details. I don't think $\cap P_i^B = M$ is necessarily true either.
Would appreciate the help.