In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until now.
I'm trying to use May-Sigurdsson's definition of Twisted $K$-theory, since the twisted $K$-homology comes more natural that way.
Actual question:
The following is an excerpt from a paper. If need be, I can also provide a ref to it. It says
where $P \to X$ is a $\mathbb C P^\infty$-principal bundle, corresponding to a twist $p \in H^3(X,\mathbb 3) \cong [X, K(\mathbb Z,3)]$ and $K$ is the $K$-theory spectrum. You can ignore the first map $K_*(\mathbb C P^\infty) \otimes K_*(P) \to K_*(P \times \mathbb C P^\infty)$, since this is "just" the Künneth theorem for (untwisted) $K$-theory. What I'm interested in is how the three maps induce maps on twisted $K$-theory, and how the maps induced by the actions are equalized by the induced map of the third map.
I've rewritten the first "equalizer by" maps and put them into a diagram.
The symbols I used are chosen to comply better with the literature (in the second part of this question), i.e. $\Pi=\mathbb C P^\infty, k=K$. Notice that the associated bundle/borel construction is morally "adding zero", since by associativity we can cancel a term in the top row in the the left colmn and middle column again. Anyway, The definition the author uses for twisted $K$-theory is as follows:
This suggests to me that when the author says "which are equalized by ..." he means to use that $K^-(-)$ is a functor between the respective categories, also respecting the maps $\bar{\alpha},\bar{\beta}$ over $\alpha, \beta$ or is there another reason for this? Going by this notation would lead us to plug the pairs of arrows $(\bar{\alpha},\alpha)$ and $(\bar{\beta},\beta)$ into the functor, which translates to
Understanding the twisted K-theory functor:
As for the more general part of the question, here is link to the book by May and Sigurdsson. Given the question, we are mainly interested in K-homology. The homology theories are defined in 20.2.4. This is for $X$ (and $J$) a param. spectrum and is furthermore the reduced case, so we have to adjoin a point (by the proof of 20.1.5.) to get to the unreduced version, and apply the suspension functor. In total, this reads as
$$ J_n(X) = \pi_n(r_!(J \wedge_B \sum\nolimits_B^\infty X_+)) $$
Using Definition 22.1.1 (with $G=e$, $\Pi=\Gamma$), we find ourselfs in the slice category over $BK(\mathbb Z, 2) \cong BK(\mathbb Z, 3)$. Now restricting to $\Pi$-principal bundles, a morphism between objects is just a continous map between the base spaces $f: X \to Y$, such that the "projection" maps (=classifying maps/twistings) commute. This "domain" category is worded simularily in Wang's paper on arxiv [p. 9] (notice that here $K$-groups = $K$-cohomology is discussed, but the category should be the same).
The lower triangle is in said category, and the upper square is what I'm interested in (sort of). Curiously, these are not $\Pi$-maps (?), and thus the upper square is not in the category of $\Pi$-principal bundles. This would at least ensure that the map $\bar{f}$ determines the lower map, according to Husemoller/Fibre bundles (p. 43). The problems persists when we apply the associated bundle construction (this is 15.4. in the book)
Because for one: Husemoller requires (p. 45) the morphisms to "come" from principal bundle morphisms, and on the other hand: The second action map with component $\mathbb C P^\infty \times K \to K$ can not be of this shape/come from a principal bundle map. Notice that in the last diagram the nodes in the top row used 22.1.4 from the book. So the question is how an arbitrary function in the top row relates to $f$ and how twisted $K$-theory behaves with respect to it. The top row is a map of spectra, and at least in the unparametrized case, maps of spectra induce natural transformations between the respective (co-)homology theories (e.g. 8.39, Switzer).
Two additional sources on parametrized spectra:
May's talk titled "What are parametrized spectra good for?" (p. 16, 30, 37 etc.)