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A plan might be to develop F. Dorais answer here via theorem 5.42 in Rosicky/Adamek’s LPAC.
Mike: Beke, Rosicky article, which you “added” is, with (full publication details, not only arXiv!) already up in the page for a while. Also some other Lieberman’s articles (preceding the newer ones you listed) on the categorical point of view are also in the list above your remark.
Thanks!
Added recent references using topos theoretic methods in model theory of AECs:
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