= 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 $p_*=0$ 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>.
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