Phantom maps vanish on represented homology

For a phantom map of spectra $p:D\to E$ and a finite CW complex $K$, Spanier-Whitehead duality identifies $D_k(K)$ with maps from the finite spectrum ${\Sigma }^k({\Sigma }^{\infty }K{)}^{\vee }$. Hence $p_{\ast }=0$ there. An arbitrary CW complex is the filtered union of its finite subcomplexes, and represented homology theory commutes with that filtered union because spheres are compact. Thus $p$ induces zero on the entire represented homology theory.