Let O G be the category of canonical orbits of G. The objects of O G are the orbit spaces G/H, H a subgroup of G, and the morphisms are the equivariant maps between them, where G acts on G/H by left multiplication. By a coefficient system for G we mean a contravariant functor from O a into the category of abelian groups. A contravariant functor from O a into the category of q-vec tor spaces will be called a rational coefficient system. The coefficient systems of either sort and the natural transformations between them form abelian categories with sufficiently many projectives and injectives [1, 1.10]. For a G-space X, the coefficient systems n n (X) and -//n (X) can be defined by _~, (X) (G/H) = _n, (Xn), _/J, (X) (G/H) = H, (Xn), n > 1, where H, ( ) denotes the singular homology with Z-coefficients. By naturality of the Hurewicz homomorphism, we get a morphism of coefficient systems h,: _n, (X) -~ Un (X), which we shall also call the Hurewicz homomorphism.