Viewing noncommutative Hopf algebras as algebras of functions on (not explicitly defined) quantum groups, (normal) Hopf algebra homomorphisms become the duals of morphisms of quantum groups. One of the more important examples is the quantum Froebenius morphism for quantizations of function algebras on semisimple Lie groups, defined by De Concini and Lyubashenko. In the first part of this talk I will discuss some of the abstract representation theory arising from multiparametrizations of this map. In the second part (as time permits) I will discuss existence/nonexistence of normal Hopf subalgebras in low Gelfand-Kirillov dimension.