It is known (cf. e.g. Theorem C.2.2.13 in the Elephant) that any (locally small) cocomplete quasitopos with a strong generator is locally presentable. In this question I am interested in cocomplete quasitoposes with just ordinary (not necessarily strong) generators, and in particular in cocomplete quasitoposes which are *not* locally presentable. Specifically, I am wondering what examples there are of such quasitoposes (even better if they are cowellpowered or at least epi-cocomplete, i.e. have arbitrary cointersections of epimorphisms). I suppose I could also look through the nLab page on quasitoposes and try to verify which of the examples satisfy my assumptions, but I thought it would be fruitful to also ask for such examples here.



Cocomplete quasitoposes that are not locally presentable
The category of [pseudotopological spaces](https://ncatlab.org/nlab/show/pseudotopological+spaces) should be a good example. It is a locally small complete and cocomplete quasitopos (indeed, it is topological over Set), the one-point space is a non-strong generator, and it is surely not locally presentable (although proving formally that it *isn't* locally presentable might be tricky). I haven't checked carefully but I suspect its epimorphisms are the maps that are surjective on underlying sets, in which case it would also be cowellpowered since any set has only a small number of quotients and any of those quotients has only a small number of pseudotopologies.


