  The notion of a quantum family of maps has been introduced in the framework
of C*-algebras. As in the classical case, one may consider a quantum family of
maps preserving additional structures (e.g. quantum family of maps preserving a
state). In this paper we define a quantum family of homomorphisms of locally
compact quantum groups. Roughly speaking, we show that such a family is
classical. The purely algebraic counterpart of the discussed notion, i.e. a
quantum family of homomorphisms of Hopf algebras, is introduced and the
algebraic counterpart of the aforementioned result is proved. Moreover, we show
that a quantum family of homomorphisms of Hopf algebras is consistent with the
counits and coinverses of the given Hopf algebras. We compare our concept with
weak coactions introduced by Andruskiewitsch and we apply it to the analysis of
adjoint coaction.
