Let $T:\mathbf{Top}_*\to \mathbf{Top}_*$ be a continuous functor and $E$ a spectrum with maps $\sigma_n:E_n\wedge S^1\to E_{n+1}$. We have a new spectrum $TE$ with structure maps $$(TE_n)\wedge S^1\stackrel{\alpha_{E_n,S^1}}{\to} T(E_n\wedge S^1)\stackrel{T\sigma_n}{\to} TE_{n+1},$$ where $\alpha_{X,Y}$ is the adjoint of $Y\to (X\wedge Y)^X\to T(X\wedge Y)^{TX}$. On the other hand, we could have considered first the spectrum $E\wedge X$ and applied the continuous functor $T$ to it in the same way. We should get a morphism $$(TE)\wedge X\to T(E\wedge X)$$ of spectra. I try to see if this is a stable equivalence for CW complexes $X$.

In order to prove this, I would like to compare two homology theories: We have a homology theory by $$h_n(X;TE) := \pi_n((TE)\wedge X).$$ and on the other hand, a functorial assigment $$X\mapsto \pi_n(T(E\wedge X)).$$ Is this also a homology theory? If so, they clearly coincide on $S^0$, so we could conclude that they coincide for all CW complexes, giving the original statement.

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    $\begingroup$ Yes. I think that someone else can give better details than I, but the key point is that your functor $T:Spectra\to Spectra$ takes homotopy pushout squares to homotopy pushout squares. A quick way to see that it commutes with suspension is to use the "shift" model for suspension of a spectrum, where $(\Sigma E)_n=E_{n+1}$. $\endgroup$ – Tom Goodwillie Oct 5 '19 at 13:19

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