Main objects in function field arithmetic are Anderson A-modules and their A-motives and A-comotives (dual A-motives). Of particular interest are those which are "abelian", since their A- motives are finitely generated, and those which are "coabelian", since their A-comotives are finitely generated. U. Hartl showed that for an Anderson A-module that is abelian and coabelian, there is a perfect pairing between its motive and its comotive, and hence an isomorphism between the dual of the motive, and the comotive. The pairing, however, is not given explicitly, and Hartl asks for such an explicit description. It also remained open whether abelian A-modules are the same as coabelian A-modules. After introducing the main objects, we will explain the main tasks. We will show that indeed abelian and coabelian are equivalent notions, and provide an explicit pairing between the A-motive and the A-comotive when the Anderson A-module is abelian/coabelian. Due to the nature of the A-motive and A-comotive, we will deal a lot with non-commutative polynomial rings and non-commutative Laurent series fields.