additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
(also nonabelian homological algebra)
Not every abelian category is a concrete category, such as Ab or $R$Mod, hence its objects do not necessarily have underlying sets whose elements one can reason about. To circumvent this and reason by “diagram chases” of elements in general abelian categories, one can instead use generalized elements in a suitable way.
This is closely related to embedding the abelian category fully-faithfully and exactly into a concrete abelian category. See at abelian category the section Embedding theorems for more on this.
One definition goes likes this (MacLane, VIII.4 and also for instance Gelfand-Manin, II.5):
An element of an object $W$ in a given abelian category $\mathcal{A}$ is an equivalence class $[X,h]$ of pairs $(X,h)$ where $X$ is an object of $A$ and $h:X\to X'$ a morphism (hence a generalized element) and the equivalence is defined as follows: $[X,h] = [Y,k]$ iff there exists an object $Z$ in $A$ and epimorphisms $u:Z\to X$, $v:Z\to Y$ such that $h\circ u = k\circ v : Z\to X'$.
However, beware that the passage to equivalence classes does not respect the abelian group structure and hence generalized elements in this sense cannot be added or subtracted. A more natural approach is discussed in (Bergman) where the actual generalized elements are remembered but a refinement of their domain is allowed, much as familiar from topos theory.
Equivalence classes of generalized elements are considered for instance in
Genuine generalized elements are considered in