Formally, is the category whose objects are those of and whose morphisms from X to Y are given by the direct limit (of abelian groups)
where the limit is taken over subobjects and such that and . (Here, and denote quotient objects computed in .) These pairs of subobjects are ordered by .
Composition of morphisms in is induced by the universal property of the direct limit.
The canonical functor sends an object X to itself and a morphism to the corresponding element of the direct limit with X′ = X and Y′ = 0.
An alternative, equivalent construction of the quotient category uses what is called a "calculus of fractions" to define the morphisms of . Here, one starts with the class of those morphisms in whose kernel and cokernel both belong to . This is a multiplicative system in the sense of Gabriel-Zisman, and one can localize the category at the system to obtain .[1]
N. Popesco; P. Gabriel (1964). "Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes". Comptes Rendus de l'Académie des Sciences. 258: 4188–4190.