Let $\mathcal{C}$ be a category. Moreover, let $\left\{U\right\}=\left\{U_{i}\right\}_{{i\in I}}$ be a family of objects of $\mathcal{C}$. The family $\left\{U\right\}$ is said to be a family of generators of the category $\mathcal{C}$ if for any object $A$ of $\mathcal{C}$ and any subobject $B$ of $A$, distinct from $A$, there is at least an index $i\in I$, and a morphism, $u:U_{i}\to A$, that cannot be factorized through the canonical injection $i:B\to A$. Then, an object $U$ of $\mathcal{C}$ is said to be a generator of the category $\mathcal{C}$ provided that $U$ belongs to the family of generators $\left\{U_{i}\right\}_{{i\in I}}$ of $\mathcal{C}$ ([4]).

By duality, that is, by simply reversing all arrows in the above definition one obtains the notion of a family of cogenerators $\left\{U^{*}\right\}$ of the same category $\mathcal{C}$, and also the notion of cogenerator $U^{*}$ of $\mathcal{C}$, if all of the required, reverse arrows exist. Notably, in a groupoid– regarded as a small category with all its morphisms invertible– this is always possible, and thus a groupoid can always be cogenerated via duality. Moreover, any generator in the dual category $\mathcal{C}^{{op}}$ is a cogenerator of $\mathcal{C}$.

1. 1.

   (Ab3). Let us recall that an Abelian category $\mathcal{A}b$ is cocomplete (or an $\mathcal{A}b3$-category) if it has arbitrary direct sums.

2. 2.

   (Ab5). A cocomplete Abelian category $\mathcal{A}b$ is said to be an $\mathcal{A}b5$-category if for any directed family $\left\{A_{i}\right\}_{{i\in I}}$ of subobjects of $\mathcal{A}$, and for any subobject $B$ of $\mathcal{A}$, the following equation holds

   $(\sum_{{i\in I}}A_{i})\bigcap B=\sum_{{i\in I}}(A_{i}\bigcap B).$

As an example consider the category $\mathcal{\mathcal{A}}b$ of Abelian groups such that if $\left\{X_{i}\right\}_{{i\in I}}$ is a family of abelian groups, then a direct product $\Pi$ is defined by the Cartesian product $\Pi_{i}(X_{i})$ with addition defined by the rule: $(x_{i})+(y_{i})=(x_{i}+y_{i})$. One then defines a projection $\rho:\Pi_{i}(X_{i})\rightarrow X_{i}$ given by $p_{i}((x_{i}))=x_{i}$. A direct sum is obtained by taking the appropriate subgroup consisting of all elements $(x_{i})$ such that $x_{i}=0$ for all but a finite number of indices $i$. Then one also defines a structural injection , and it is straightforward to prove that $\mathcal{\mathcal{A}}b$ is an $\mathcal{\mathcal{A}}b6$ and $\mathcal{\mathcal{A}}b4^{*}$ category. (viz. p 61 in ref. [4]).