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I am currently reading Part 3 of Adams's book "stable homotopy and generalized cohomology", and I got stuck when following his argument.

In Proposition 5.4, he states that when $W$ and $X$ are finite spectra, then $S$-dual map $[W, Z \wedge X^{*}] \to [W \wedge X, Z]$ is an isomorphism. Adams claims that this can be proved by passing from the case that Z is a finite spectra, and I don't understand here.

It's okay that this is an isomorphism when $Z$ is finite, and I would like to pass to colimits. I think that every CW-spectrum can be represented by a union of finite spectra because for every stable cells we can take finite spectra which contains the stable cell. My question is,

(1) If $Z = \cup Z_{\alpha}$, is it true that $Z \wedge X^{*} = \cup (Z_{\alpha} \wedge X^{*})$ ? Can we deduce this result from formal properties of smash products? (I haven't read Adams's construction of smash products.) I think that $\cup Z_{\alpha}$ is a filtered colimit in the category of CW-spectra (notice that we don't take homotopy classes of maps). Is $\cup Z_{\alpha}$ also a filtered colimit in the stable homotopy category (the category which has its morphism as the homotopy classes of the maps in the category of CW-spectra)? I've heard that taking smash product functor is left adjoint to some functor, so I would like to use this fact, although I am still wondering whether inclusion maps still remains inclusion after taking smash product.

(2) If $W$ and $X$ are finite, is $W \wedge X$ also finite? Can we deduce this without looking its construction?

An argument like above appears frequently in his book, so I have some troubles in understanding his thoughts.

Thank you very much.

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  • $\begingroup$ and I don't know whether I should post question something like this to stackexchange or overflow... $\endgroup$ – Tominaga May 24 at 8:24
  • $\begingroup$ Note: I've found a solution for the second question and I am wondering that an appendix in Ravenel's orange book would give me the solution. I will write an answer when I get a clear understanding. $\endgroup$ – Tominaga May 27 at 5:36

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