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For any space/spectrum $X$ one can define the associated tower of Postnikov sections $\{ P^n X\}_{n}$ as a Bousfield localization with respect to all spheres with dimension $>n.$ Therefore, we have the following properties:

(1) $X \to P^nX$ is the localization map.

(2) $\pi_k(P^n(X))=0$ for $k >n$.

(3) $\pi_k(X) \to \pi_k(P^n(X))$ is an isomorphism for $k \leq n.$

Question: For any two spaces/spectra $X$ and $Y,$ is there any canonical map/ functorial map from $P^n(X) \wedge P^n Y \to P^n(X\wedge Y)?$ Mainly, I am concerned about symmetric spectra.

Also, is it true that $P^n(P^n(X) \wedge P^n(Y))\cong P^n(X \wedge Y)?$

Thank you so much in advance. Any help will be appreciated.

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