 Further, if the family formula_59 is directed (i.e.for any two objects in the family, there is a third object in the family that contains the two), and formula_1 is another subobject of formula_17, we have

Grothendieck categories are well-powered (sometimes called "locally small", although that term is also used for a different concept), i.e.the collection of subobjects of any given object forms a set (rather than a proper class).