	Let "G" be a finite group (in fact everything will work verbatim for a profinite group). Then for any two finite "G"-sets "X" and "Y" we can define an equivalence relation among spans of "G"-sets of the form formula_1 where two spans formula_1 and formula_3are equivalent if and only if there is a "G"-equivariant bijection of "U" and "W" commuting with the projection maps to "X" and "Y". This set of equivalence classes form naturally a monoid under disjoint union; we indicate with formula_4 the group completion of that monoid. Taking pullbacks induces natural maps formula_5.