  The notion of families of quantum invertible maps ($C^*$-algebra
homomorphisms satisfying Podle\'s condition) is employed to strengthen and
reinterpret several results concerning universal quantum groups acting on
finite quantum spaces. In particular Wang's quantum automorphism groups are
shown to be universal with respect to quantum families of invertible maps.
Further the construction of the Hopf image of Banica and Bichon is phrased in
the purely analytic language and employed to define the quantum subgroup
generated by a family of quantum subgroups or more generally a family of
quantum invertible maps.
