= Solution
Let $\operatorname{Map}_{\mathrm{sp}}(D,E)$ be the space of point-set <maps of topological spectra>. Its underlying set is the first set in the comparison. A member is a compatible collection $D_j\to E_j$, rather than independently chosen maps at the levels.
Because the source is cellular and the target is an <Omega-spectrum>, taking <spectrum homotopy> classes gives all maps in the <stable homotopy category>:
$$
\boxed{[D,E]_{\mathrm{st}}=\pi_0\operatorname{Map}_{\mathrm{sp}}(D,E).}
$$
Thus the first set maps onto the second. \b[Two point-set maps have the same stable class precisely when they are spectrum-homotopic], using the cellular cylinder and fibrant target. The homotopy must commute with the structure maps throughout; unrelated levelwise homotopies are not the definition.
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