The main theorem of this paper describes an isomorphism between the Witt-Burnside ring over a commutative monoid ring \({\mathbb Z}[M]\) and the Grothendieck ring of a category whose objects are almost finite \(G\)-sets \(X\) for a profinite group \(G\), equipped with a map \(X\rightarrow M\), that is constant on \(G\)-orbits. As a consequence, the author generalizes several well known maps on Witt vectors to homomorphisms between Witt-Burnside rings. The result forms the basis for a new construction of Witt-Burnside rings.