Abstract
Under GCH, a set functor F does not preserve finite unions of non-empty sets if and only if the category Coalg F of all F-coalgebras is universal. Independently of GCH, we show that for any non-accessible functor F preserving intersections, the category Coalg F has a large discrete full subcategory, and we give an example of a category of F-coalgebras that is not universal, yet has a large discrete full subcategory.