If one accepts the notion of subcategory without any qualification (as discussed there), then:
A subcategory of a category is a full subcategory if for any and in , every morphism in is also in (that is, the inclusion functor is full).
This inclusion functor is often called a full embedding or a full inclusion.
Notice that to specify a full subcategory of , it is enough to say which objects belong to . Then must consist of all morphisms whose source and target belong to (and no others). One speaks of the full subcategory on a given set of objects.