On quantum maps into quantum semigroups

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.