Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory from the category of models of that theory, for example, the theory of groups from its category of models. There are other related dualities corresponding to other 'doctrines', see A Duality Relative to a Limit Doctrine.
This idea of recovering something from its category of models seems to resemble closely the Tannakian reconstruction of an algebraic entity from its category of modules or comodules or whatever.
So I was wondering whether there is a way to see doctrinal, Gabriel-Ulmer style reconstruction and Tannaka style reconstruction as instances of something more general.
The only explicit reference I have found is Brian Day in Enriched Tannaka reconstruction writing "Our approach provides a synthesis of Tannaka reconstruction and Gabriel-Ulmer duality," but I haven't found the paper very helpful.
I've posed this question at greater length here.
