On simple classes of special generic maps and round fold maps and fold maps obtained by composing projections

Fold maps are fundamental tools in the theory of singularities of differentiable maps and its applications to geometry. They are higher dimensional variants of Morse functions. Classes of special generic maps and round fold maps are important classes of fold maps. {\it Special generic} maps are higher dimensional variants of Morse functions on homotopy spheres with exactly two {\it singular points}: canonical projections of unit spheres are special generic. Round fold maps are Morse functions obtained as doubles of Morse functions, or fold maps such that the set of all the singular points are embeddings and that the images are concentric. In the present paper, we discuss compositions of these maps with canonical projections. For example, we observe that these compositions for special generic maps of simple classes are regarded as round fold maps in considerable cases. We also present round fold maps we cannot represent in this way, seeming to be represented so. Note that such compositions are natural operations in related theory of differentiable maps.