Representations, duals and quantum doubles of monoidal categories

summary:[For the entire collection see Zbl 0742.00067.]\par The Tanaka-Krein type equivalence between Hopf algebras and functored monoidal categories provides the heuristic strategy of this paper. The author introduces the notion of a double cross product of monoidal categories as a generalization of double cross product of Hopf algebras, and explains some of the motivation from physics (the representation theory for double quantum groups).\par The Hopf algebra constructions are formulated in terms of monoidal categories $\underline C$ and functors $\underline C\to\underline{\text{Vec}}$ (finite-dimensional vector spaces) and generalized by replacing $\underline{\text{Vec}}$ by another monoidal category $\underline V$. It is interesting to remark that the monoidal category $\underline C$ (functored over a category $\underline V$) has a ``Hopf algebra like structure'', $\underline V$ having the role of the ground field. If $\underline V$ is a quasitensor category, a coadjo