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I guess we could also say FinCat has enough discrete objects, in that every object admits an essentially surjective functor from a finite set.
There are some papers by Almeida and Weil, e.g. Profinite categories and semidirect products 1, JPAA vol 123, 1998, p. 1-50, that may be relevant. A similar subject is handled in J. P. Jones, Profinite categories, implicit operations and pseudovarieties of categories , again JPAA, vol. 107, 1996. I have not checked if all the terms used there have exactly the same meaning as you are intending, David.
It is perhaps worth noting that way back in 1980 Dominique Bourn and Jean-Marc Cordier, worked on Distributeurs et la Theorie de la Forme (i.e. abstract shape theory) and used codensity monads in the bicategory Dist. That is in CTDG 21, 161-189. Again I’m not sure if that helps, but it is in the same general area (and Shape Theory seems to be lurking around in some of the stuff being discussed in the Café as well).
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