Is there a standard way to construct the shift map on an infinite product or coproduct of a direct or inverse system of spectra that induces the standard shift map of abelian groups in homology? Is it constructed differently for the product and coproduct? If there is some standard construction(s), what is it?
Can we just assume that we are (up to homotopy) dealing with $\Omega$-spectra which have some natural group structure?
Thanks.