  We define the universal quantum group $\mathcal{H}$ that preserves a pair of
Hopf comodule maps, whose underlying vector space maps are preregular forms
defined on dual vector spaces. This generalizes the construction of Bichon and
Dubois-Violette (2013), where the target of these comodule maps are the ground
field. We also recover the quantum groups introduced by Dubois-Violette and
Launer (1990), by Takeuchi (1990), by Artin, Schelter, and Tate (1991), and by
Mrozinski (2014), via our construction. As a consequence, we obtain an explicit
presentation of a universal quantum group that coacts simultaneously on a pair
of $N$-Koszul Artin-Schelter regular algebras with arbitrary quantum
determinant.
