nLab
element in an abelian category

An element of an object $W$ in a fixed abelian category $A$ is an equivalence class $[X,h]$ of pairs $(X,h)$ where $X$ is an object of $A$ and $h:X\to X\prime$ 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 $h\circ u=k\circ v:Z\to X\prime$.

Notice that subtlety in the equivalence relation.

Elements in an abelian category can be used for diagram chasing?.

• Sergei Gelfand, Yuri Manin, Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original. Springer 1996. xviii+372 pp.; 2nd corrected ed. 2002.Gelfand-Manin?.