= Solution
A stable map $u:D\to E$ induces a degree-preserving additive <natural transformation> of <represented cohomology theories> by postcomposition with $\Sigma^q u$ in degree $q$. This transformation commutes with the suspension isomorphisms. The resulting map
$$
[D,E]_{\mathrm{st}}\longrightarrow\operatorname{Nat}_{\Sigma}(\widetilde D^*,\widetilde E^*)
$$
is onto, but need not be one-to-one. Its kernel consists of <hyperphantom maps of spectra>: maps $u$ for which $uv=0$ for every stable map
$$
v:\Sigma^{-j}\Sigma^\infty X\longrightarrow D,
$$
for every <based space> $X$ and every $j\ge0$. Equivalently, they induce zero in all degrees of <represented cohomology theory> on all <based spaces>. Therefore
$$
\boxed{\operatorname{Nat}_{\Sigma}(\widetilde D^*,\widetilde E^*)\cong[D,E]_{\mathrm{st}}/\operatorname{HPh}(D,E).}
$$
This condition tests arbitrary <suspension spectra> with negative shifts. Merely testing <finite spectra> defines a <phantom map of spectra>, which is a weaker condition. In particular the homology conclusion in Question 5 should not be substituted for the cohomology condition here.
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