What is a locally cosmall category relative to a universe?
Unfortunately, there is a clash of terminology: Some people call a category locally small if the hom-classes are sets. These people call a category well-powered if the subobjects of any object form a set; dually, co-well-powered refers to quotients. For these people, there is no notion of locally cosmall. Other people (and this seems to be the context of the question) call a category locally small if the subobjects of any object form a set (i.e. what the former call well-powered), and the dual notion concerning quotients locally cosmall (i.e. what the former call co-well-powered). For a reference, see Pareigis' "Categories and functors", Section 1.6. If you work with universes, replace "set" by "element of U" and "class" with "subset of U".