Definition 0.1   Let $\mathcal{A}$ be an Abelian cocomplete category, defined as the dual of an Abelian complete category.

A $C_3$ -category is defined as a cocomplete Abelian category $\mathcal{A}$ such that the following distributivity relation holds for any direct family $\left\{A_i\right\}$ and any subobject $B$ :

$$(\bigcup A_i) \bigcap B = \bigcup (A_i \bigcap B),$$ ([1])

Remark 0.1  

A $C_3$ -category is also called an $\mathcal{A}b5$ -category.

Example 0.1   The dual of the Cartesian closed category of finite Abelian quantum groups with exponential elements (including Lie groups) and quantum group homomorphisms is a $C_3$ -category.