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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?