??so what about idea of something sort of like ... ??... "getting the objects from presheaves on the site where the morphisms are just the localizations, but getting the morphisms from presheaves on the site where the morphisms are more general maps" .?? ... ???and so forth ... ???...

??any model/formula confusion here?? ... ???or something ?? ... ??and/or some sort of level confusion ... ??something about stack vs sheaf here .... ????and so forth ??? .....