Thanks. I had interpreted Theorem 8 as showing total categories to be equivalent to compact categories, but this is not true, because Ulmer's "Yoneda embedding" is really a restricted variant.

@MartinBrandenburg: compact categories are cocomplete and complete (by virtue of Ulmer's theorem), so it seems reasonable to also assume cocompleteness for strongly compact categories (it also seems a natural condition to impose to ask for preservation of small colimits from $\mathcal C$). Ulmer requires small-cocompleteness, for instance. I don't know whether the same results will hold if you relax that assumption.

@MartinBrandenburg: I did indeed, thanks :)

I've updated my answer to correct the mistake Ivan pointed out.

@IvanDiLiberti: I overlooked that subtlety. Thanks. This isn't quite the right concept.

Isn't this a straightforward exercise in dualisation? A cotopos is simply a category whose opposite is a topos. With which part are you having difficulty?

@MartinBrandenburg: it is true that the free coproduct completion may have been discovered independently many times. However, the consistent terminology in these early references (namely, "the category of families") suggests there is a common ancestor, as I don't think this terminology is inevitable. It could be that there isn't a good earliest reference. However, I would rather cite an original reference if one exists.