What Eduardo means by "Giraud topos" is a category validating all conditions of the Giraud theorem with the exception of having a small generating family. These guys can be elementary toposes or not. There is Freyd's example where objects are set X with an ordinal indexed family of bijections from X to X and morphisms are maps commuting with all these ordinal many maps. This is a locally small cocomplete elementary topos lacking a s mall generating family. In 2011 on this list Johnstone gave the example of a category which superficially is like Freyd's example but the class many endomaps are not required to be bijections. This category is also bicomplete and locally small but it hasn't got a subobject classifier. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]