  A direct relation between the enumeration of ordinary maps and that of fully
simple maps first appeared in the work of the first and last authors. The
relation is via monotone Hurwitz numbers and was originally proved using
Weingarten calculus for matrix integrals. The goal of this paper is to present
two independent proofs that are purely combinatorial and generalise in various
directions, such as to the setting of stuffed maps and hypermaps. The main
motivation to understand the relation between ordinary and fully simple maps is
the fact that it could shed light on fundamental, yet still not
well-understood, problems in free probability and topological recursion.
