Suppose I work in the completeley naive homotopy category of spectra, by which I mean sequences $E = (E_n)_{n = 0, 1, \dots}$ together with maps $\sigma_{E,n}: S^1 \wedge E_n \to E_{n+1}.$ We might require the $E_n$ to be CW complexes and the $\sigma_{E,n}$ to be homeomorphisms onto subcomplexes. This has a naive notion of weak equivalence, and so on, hence models stabel homotopy theory (except that the point set level category is not very nice). This is basically the approach described by Adams (going back to Bordman, I suppose).
Now a crucial additional part of structure is that the stable homotopy category admits a symmetric monoidal structure $\wedge$ which extends the smash product on spaces (which map to the stable homotopy category via the suspension spectrum). This is really hard to construct; Adams spends some thirty pages on it, and as far as I understand most of the modern, complicated point set level categories are precisely designed to afford a nice smash product.
However, suppose $E = (E_n)$ and $F = (F_n)$ are two spectra. Define a new spectrum $G = (G_n)$ as follows: $$G_{2n} = E_n \wedge F_n,$$ $$G_{2n+1} = S^1 \wedge E_n \wedge F_n,$$ and $\sigma_{G, 2n}$ is the natural isomorphism, whereas $\sigma_{G, 2n+1}: S^1 \wedge S^1 \wedge E_n \wedge F_n \to S^1 \wedge E_n \wedge S^1 \wedge F_n \to E_{n+1} \wedge F_{n+1}$ combines the switch homeomorphism and both structure maps. This construction is symmetric, extends the smash product on spaces, but not obviously associative (on the homotopy level). I think $G$ affords a natural map to $E \wedge F$ as constructed by Adams.
Question: Is this map a weak equivalence?
My intuition would be that $G'_n = X_n \wedge Y_n$ would define a "$S^2$-spectrum", i.e. an object in the category of "spaces with $S^2$ inverted". But inverting $S^2$ should (up to homotopy) be the same as inverting $S^1,$ and the above construction would be my guess to realise this equivalence.
