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.