and their duals AB3*) For every indexed family (Ai) of objects of A, the product PAi exists in A (i.e. A is complete). AB4*) A satisfies AB3*), and the product of a family of epimorphisms is an epimorphism. AB5*) A satisfies AB3*), and filtered limits of exact sequences are exact. Axioms AB1) and AB2) were also given. They are what make an additive category abelian. Specifically: AB1) Every morphism has a kernel and a cokernel. AB2) For every morphism f, the canonical morphism from coim f to im f is an isomorphism. Grothendieck also gave axioms AB6) and AB6*). AB6) A satisfies AB3), and given a family of filtered categories and maps , we have , where lim denotes the filtered colimit. AB6*) A satisfies AB3*), and given a family of cofiltered categories and maps , we have , where lim denotes the cofiltered limit. Elementary properties Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group. Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category. In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage of f, while the monomorphism is called the image of f. Subobjects and quotient objects are well-behaved in abelian categories. For example, the poset of subobjects of any given object A is a bounded lattice. Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A. The abelian category is also a comodule; Hom(G,A) can be interpreted as an object of A. If A is complete, then we can remove the requirement that G be finitely generated; most generally, we can form finitary enriched limits in A. Related concepts Abelian categories are the most general setting for homological algebra. All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and derived functors. Important theorems that apply in all abelian categories include the five lemma (and the short five lemma