 Solution

Let ${\mathrm{Map}}_{\mathrm{sp}}(D,E)$ 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:Thus the first set maps onto the second. 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.