Partial monoids and Dold-Thom functors

Dold-Thom functors are generalizations of infinite symmetric products, where integer multiplicities of points are replaced by composable elements of a partial abelian monoid. It is well-known that for any connective homology theory, the machinery of $\Gamma$-spaces produces the corresponding linear Dold-Thom functor. In this note we construct such functors directly from spectra by exhibiting a partial monoid corresponding to a spectrum.