One often extends this notion to a cogenerating family of objects, which is a (usually small) set of objects in such that the family is jointly faithful. This means that for any pair , if they are indistinguishable by morphisms to in the sense that
Much more generally, we have
Every topos with a small set of generators (e.g., a well-pointed topos, or a Grothendieck topos), and that has products of objects indexed over sets no larger in cardinality than the generating set, admits an injective cogenerator.
Let be a set of generators for the topos; as usual, let be the subobject classifier. We claim that a product
is a cogenerator. For suppose are distinct morphisms. The contravariant power object functor is faithful (a familiar fact, since it is monadic), so that are distinct. Since the objects form a generating set, there is some such that the composites
are distinct. The map may be transformed to a map , and it follows that the two composites
are distinct. For any other , we may uniformly define to be the map classifying the maximal subobject of , so that these maps together with collectively induce a map
that yields distinct results when composed with and . This proves the claim.
The object is injective because already is injective (see Mac Lane-Moerdijk, IV.10), and it is a general fact that in a cartesian closed category (or more generally a closed monoidal category), an exponential (or internal Hom) whose base is injective is also injective, and products of injective objects are injective.
Notice also that the existence of a small (co)generating family is one of the conditions in one version of the adjoint functor theorem. We may conclude, for example, that Grothendieck toposes are cototal (q.v.).
The concept of cogenerator is dual to that of separator, so it can also be referred to as a coseparator.