TITLE: Comparison: Formal Wirthmüller isomorphism of Fausk-Hu-May vs. Balmer et. al
QUESTION [7 upvotes]: $\newcommand{\Cc}{\mathcal{C}}$
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This question is about comparing the approaches for a formal Wirthmüller isomorphism by Fausk-Hu-May [FHM] in isomorphisms between left and right adjoints and by Balmer-Dell'Ambrogio-Sanders [BDS] in Grothendieck-Neeman duality and the Wirthmüller isomorphism.
In both articles, a formal Wirthmüller isomorphism is studied: we are given a tensor-exact functor $f^*: \Dd \to \Cc$ between tensor-triangulated categories that has both a left adjoint $f_!: \Cc \to \Dd$ and a right adjoint $f_*: \Cc \to \Dd$ and we are looking for a `twisted isomorphism' between $f_!$ and $f_*$. In [FHM] this takes the form of an isomorphism $f_*(-) \overset{\cong}{\implies} f_!(- \otimes C)$ for some object $C \in \Cc$, and in [BDS] this takes the form of an isomorphism $f_*(- \tensor \omega_f) \overset{\cong}{\implies} f_!(-)$ for another object $\omega_f \in \Cc$.
In [BDS] a comparison is given with the approach of [FHM]: it is stated in proposition 4.4 that if the situations of [BDS] and [FHM] both apply simultaneously, then the object $\omega_f$ is the dual in $\Cc$ of the object $C$. However, I don't understand the rather short proof: it is claimed that the formal assumptions already give an isomorphism $f_* \cong f_!(- \tensor C)$ by [FHM, Thm. 8.1]. But it seems to me that [FHM, Thm. 8.1] requires some non-formal input, namely one needs to explicitly check that $f_*(G) 
\to f_!(G \tensor C)$ is an isomorphism for some set of generating compact dualizable objects $\{G\}$ of $\Cc$. Indeed, it takes May in the follow-up article The Wirthmüller isomorphism revisited quite some time to prove that the latter condition is satisfied in the setting of equivariant homotopy theory. It seems that Proposition 4.4 of [BDS] is cutting this short.
Question 1: Why does it follow in [BDS,Prop. 4.4] that $f_*(-) \implies f_!(- \otimes C)$ is a natural isomorphism?
Question 2: If it doesn't follow, is it otherwise possible to compare $C$ and $\omega$ without checking that $f_*(-) \implies f_!(- \otimes C)$ is a natural isomorphism on all compact generators?
Let's assume that we know that both isomorphisms hold and additionally that $\omega_f$ is invertible (with inverse $C$). Then both approaches give a natural isomorphism $f_*(-) \implies f_!(- \otimes C)$.
Question 3: (Why) do these two isomorphisms agree?

REPLY [3 votes]: Thanks for pointing this out and sorry for the confusion. Proposition 4.4 of our paper does appear to be misstated. The meaning of "Wirthmüller context in the sense of [FHM]" needs to be strengthened to include the isomorphism that [FHM, Theorem 8.1] doesn't provide for free. The Proposition could be (correctly) stated as: "Suppose the basic adjunction $f^* \dashv f_*$ satisfies a Wirthmüller isomorphism in the sense of [FHM]; i.e., suppose that $f^*$ has a left adjoint $f_!$ and that there exists an object $C \in \mathcal C$ together with an isomorphism $f_!(C\otimes -)\simeq f_*(-)$. Then its dual $\mathrm{hom}(C,1)$ is isomorphic to $\omega_f$."
The point is that we can recognize the dualizing object from existing isomorphisms that have been established in specific examples by specific non-formal means. (See e.g., Remark 2.16 of my other paper, "The compactness locus of a geometric functor and the formal construction of the Adams isomorphism".)
In any case, you are correct that there appears to be no reason why the choice of isomorphism $f_!(C)\simeq f_*(1)$ would imply an isomorphism $f_!(C\otimes -)\simeq f_*(-)$. Indeed, this is one of the points about these works --- the $C$ that appears in [FHM] is non-canonical and the isomorphism of Theorem 8.1 doesn't come for free; in contrast, the relative dualizing object is canonical and the isomorphism of [BDS] comes essentially for free (in our rigidly-compactly generated setting).
Also, the real point of our Remark 4.3 and (misstated) Proposition 4.4 is to show that the Wirthmüller isomorphism (à la [FHM]) implies the Grothendieck-Neeman isomorphism, contrary to the discussion in [FHM], about these being distinct settings. I think this is one of the things that was clarified by our paper.
Finally, as you mention (in your answer) the example from May's second paper doesn't exactly fit into our setup since the equivariant stable homotopy category over the $N$-fixed universe is not rigidly-compactly generated --- not all of the generators are dualizable. Nevertheless, one can give an example of an $f^*:\mathcal D \rightarrow \mathcal C$ between rigidly-compactly generated categories which admits an invertible object $C\in \mathcal C$ such that $f_!(C) \simeq f_*(1)$ and yet $\mathrm{hom}(C,1) \not\simeq \omega_f$. I'll give this example as an answer to your other MathOverflow question. This will thus provide an example (fitting into our setup) where the associated [FHM] map $f_* \rightarrow f_!(C\otimes -)$ is not an isomorphism.