One of the classical theorems of model theory is the Chang-Łos-Suszko preservation theorem that states that the theories formulated in FOL (first order logic) that are preserved under [direct limits][1] (generalized unions) are precisely those theories that are axiomatizable by a collection of $\Pi_2$-sentences, i.e., those of the form $\forall x_1 ...\forall x_m \exists y_1...\exists y_n ~\delta$, where $\delta$ is quantifier-free (see, e.g., [these slides][2] of Benno van den Berg).

The above motivates the following question concerning [inverse limits][3] (of inverse systems). NB: category theorists commonly refer to inverse limits as limits and direct limits as colimits.


**Question.** *Is there a characterization of FOL theories that are preserved under inverse limits*? 


  [1]: https://en.wikipedia.org/wiki/Direct_limit
  [2]: https://staff.fnwi.uva.nl/b.vandenberg3/Onderwijs/Modeltheory_2012/slides_modeltheorie_lecture4.pdf
  [3]: https://en.wikipedia.org/wiki/Inverse_limit