In detail, let A be an object of some category. Given two monomorphisms
u : S -> A and v : T -> A
with codomainA, we write u v if ufactors throughv--that is, if there exists ? : S -> T such that . The binary relation ? defined by
u ? v if and only if u v and v u
is an equivalence relation on the monomorphisms with codomain A, and the corresponding equivalence classes of these monomorphisms are the subobjects of A. (Equivalently, one can define the equivalence relation by u ? v if and only if there exists an isomorphism ? : S -> T with .)
The relation partial order on the collection of subobjects of A.
The collection of subobjects of an object may in fact be a proper class; this means that the discussion given is somewhat loose. If the subobject-collection of every object is a set, the category is called well-powered or sometimes locally small.
To get the dual concept of quotient object, replace "monomorphism" by "epimorphism" above and reverse arrows. A quotient object of A is then an equivalence class of epimorphisms with domain A.
In Set, the category of sets, a subobject of A corresponds to a subsetB of A, or rather the collection of all maps from sets equipotent to B with image exactly B. The subobject partial order of a set in Set is just its subset lattice.
Given a partially ordered classP = (P, P as objects, and a single arrow from p to q iff p q. If P has a greatest element, the subobject partial order of this greatest element will be P itself. This is in part because all arrows in such a category will be monomorphisms.