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I (only) now realize that I pretty much missed the story of familial regularity and exactness. But also it was easy to miss, with the entries that are unified by this not pointing back to it.
To rectify this I have created now a floating TOC and am including it into all the relevant entries:
Please check out that TOC and edit/modify as need be.
There is a bug in your first link, Urs, but please don't fix it; I'm reporting it to Andrew.
It would be nice if somebody can add to this contents page also links to discussions of embedding/reconstruction theorems for categories with various degrees of regularity/exactness due Barr, van Osdol and others. Unfortunately, I do not have these things clear in my head yet to do it myself.
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