The notion of the quantum space of all maps between quantum spaces was invented by Piotr M. Soltan. His pioneering work was mainly focused on finite-dimensional C*-algebras, which are matrix-algebra bundles over a finite set S. We propose a generalization of this concept that includes arbitrary compact Hausdorff spaces X (instead of finite sets S) and takes into account the topology of X. In this context, the notion of the free product of copies of a unital C*-algebra topologically indexed by a compact Hausdorff space arises naturally, and enjoys some desired functoriality. Based on joint work with Thomas Timmermann.