  We define algebraic families of (all) morphisms which are purely algebraic
analogs of quantum families of (all) maps introduced by P.M. Soltan. Also,
algebraic families of (all) isomorphisms are introduced. By using these notions
we construct two classes of Hopf-algebras which may be interpreted as the
quantum group of all maps from a finite space to a quantum group, and the
quantum group of all automorphisms of a finite NC space. As special cases three
classes of NC objects are introduced: quantum group of gauge transformations,
Pontryagin dual of a quantum group, and Galois-Hopf-algebra of an algebra
extension.
