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For cellular Omega-spectra $D,E$, natural degree-preserving additive operations $D^{\ast }\to E^{\ast }$ commuting with suspension are compatible families $[D_j,E_j{]}_{\ast }$ by the Yoneda lemma. Since $D\simeq {\mathrm{hocolim}}_j{\Sigma }^{-j}{\Sigma }^{\infty }{D}_j$, the Milnor exact sequence for maps of spectra realizes every such family. Its kernel consists precisely of hyperphantom maps of spectra. Thus operations determine stable maps only modulo that kernel.

The Yoneda lemma, applied to the representing Omega-spectrum spaces in each degree, makes the comparison more explicit. PutThe structure map of $D$ supplies $T_j\to T_{j+1}$, and $D$ is the homotopy colimit of spectra of this sequence. The inverse-tower transition on maps into $E$ is precomposition with $T_j\to T_{j+1}$. Under the suspension spectrum adjunction it takes $[D_{j+1},E_{j+1}{]}_{\ast }$ to $[D_j,E_j{]}_{\ast }$ by taking the adjoint through $D_j\to \Omega D_{j+1}$ and the Omega-spectrum identification $E_j\simeq \Omega E_{j+1}$.

Consequently the maps of represented cohomology theories are exactly the compatible degreewise operations:Only nonnegative degrees need be listed, since the suspension compatibility recovers the negative degrees. The Milnor exact sequence for maps of spectra gives the sharper descriptionThus $\mathrm{HPh}(D,E)\cong {\lim }_j^1[\Sigma D_j,E_j{]}_{\ast }$, with bonding maps coming from the same telescope. For an inverse tower $A_j$ with maps $r_j:A_{j+1}\to A_j$, $\lim A_j$ is the kernel and ${\lim }^1{A}_j$ is the cokernel ofThe first comparison quotients by spectrum homotopy; the second quotients by hyperphantom maps. These are different identifications.