  We analyze the recent examples of quantum semigroups defined by M.M. Sadr who
also brought up several open problems concerning these objects. These are
defined as quantum families of maps from finite sets to a fixed compact quantum
semigroup. We show that these are special cases of free products of quantum
semigroups. This way we can answer all the questions stated by M.M. Sadr. Along
the way we discuss the question whether restricting the comultiplication of a
compact quantum group to a unital $\mathrm{C}^*$-subalgebra defines such a
structure on the subalgebra. In the last section we show that the quantum
family of all maps from a non-classical finite quantum space to a quantum group
(even a finite classical group) might not admit any quantum group structure.
