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(This is the follow-up of [this question](https://mathoverflow.net/questions/383783/). I have repeated the motivation for the question.)


The following situation frequently arises in abstract category theory: we have symmetric closed monoidal categories $\Cc$ and $\Dd$ and a closed symmetric monoidal functor $f^\*: \Dd \to \Cc$ that admits both a left adjoint $f\_!: \Cc \to \Dd$ and a right adjoint $f\_\*: \Cc \to \Dd$, and we are interested in the relation between $f\_!$ and $f\_\*$. This is often called a [Wirthmüller context](https://ncatlab.org/nlab/show/Wirthm%C3%BCller+context), named after the inspiring example of the Wirthmüller isomorphism in equivariant homotopy theory. (The symbol $f^\*$ is just notation: there is not necessarily a morphism $f$ around.)


To compare $f\_!$ and $f\_\*$, Fausk-Hu-May [FHM] assume in the article *isomorphisms between left and right adjoints* the existence of an object $C \in \Cc$ satisfying $f\_!(C) \simeq f\_\*(1\_{\Cc})$, where $1\_{\Cc}$ denotes the monoidal unit of $\Cc$. From this data, a natural transformation $f\_\*(-) \to f\_!(- \tensor C)$ is constructed and conditions are given for when this is an isomorphism (the 'formal Wirthmüller isomorphism').


As remarked by Balmer-Dell'Ambrogio-Sanders [BDS] in the article *Grothendieck-Neeman duality and the Wirthmüller isomorphism*, the object $C$ is *a priori* not unique and it doesn't come with a `characterizing description'. My main question is:


**Question (1): Do we have counterexamples for essentially uniqueness of an object $C$ satisfying $f\_!(C) = f\_\*(1\_{\mathcal{C}})$?**


I actually want to follow [BDS] and stay in a more restricted setting. We assume that $\Cc$ and $\Dd$ are tensor-triangulated (or, if you prefer, stably symmetric monoidal stable $\infty$-categories) and that they are generated by a set of compact objects and admit internal homs (or equivalently: the tensor products commute with coproducts in both variables). Moreover, we assume that $\Cc$ and $\Dd$ are *rigid*, i.e. the compact objects are the same as the strongly dualizable objects. Finally, we assume that $f^\*: \Dd \to \Cc$ is exact and strong monoidal and that it admits both a left adjoint $f\_!: \Cc \to \Dd$ and a right adjoint $f\_\*: \Cc \to \Dd$.


Examples of rigid tensor triangulated categories are spectra $\operatorname{Sp}$, genuine $G$-spectra $\operatorname{Sp}^G$ for a compact Lie group $G$ and the derived category $D(R)$ for a ring $R$. Several more advanced examples of such categories and functors are in examples 4.6 - 4.8 in the article *Grothendieck-Neeman duality and the Wirthmüller isomorphism* by Balmer-Dell'Ambrogio-Sanders, based on 'finite group schemes', 'motivic homotopy theory' and 'cohomology rings of classifying spaces'. However, I don't really understand them and in particular I don't know if they give counterexamples to question (1) above. Hence:


**Question (2): What are simple examples of functors $f^\*: \Dd \to \Cc$ satisfying the above conditions?**


**Attempts:**
I expect that it shouldn't be hard to come up with counterexamples for (1), but somehow I cannot make them work. For example:


* Consider a finite group $G$ and let $f: H \hookrightarrow G$ be a subgroup inclusion. Then the restriction functor $f^\*: \Sp^{G} \to \Sp^H$ between (genuine) equivariant spectra is symmetric monoidal with left adjoint $f\_!(X) = G \ltimes\_H X$ and right adjoint $f\_\*(X) = F(G/H\_+,X)$. In this case we actually have $f\_! \simeq f\_\*$ by the Wirthmüller isomorphism. And I think that if $C \in \Sp^H$ satisfies $f\_!(C) \simeq f\_\*(\mathbb{S}\_H) \simeq f\_!(\mathbb{S}\_H)$, then we already have $C \simeq \mathbb{S}\_H$, so this won't give a counterexample.
* Consider a finite group $G$ and a normal subgroup $N < G$, and let $f: G \to G/N$ be the projection. Then $f^\*: \Sp^{G/N} \to \Sp^G$ between (genuine) equivariant spectra is symmetric monoidal with left adjoint $f\_!(X) = X/N$ and right adjoint $f\_\*(X) = X^N$. Now an object $C$ with $f\_!(C) \simeq f\_!(\mathbb{S}\_G)$ is no longer unique since $f\_!(\Sigma^{\infty}\_+G) \simeq f\_!(\Sigma^{\infty}\_+ G/N)$. But we don't have the Wirthmüller isomorphism $f\_! \simeq f\_\*$ so it doesn't immediately produce a counterexample.


I also tried a little outside the range of equivariant homotopy theory, but there I fail to even come up with functors satisfying the required criteria:


* Let $f: \Z \hookrightarrow \Q$ and let $f^\*: \Dd = D(\Q) \to \Cc = D(\Z)$ be the forgetful functor. This is symmetric monoidal (since $\Q$ is a (derived) localization of $\Z$). However, it does not preserve compact objects: $\Q$ is not finitely generated over $\Z$. **Edit:** as remarked in the comments, it is actually only lax symmetric monoidal, since the monoidal unit is not sent to the monoidal unit.
* Let $f^\*: \Dd = D(\Z/2\Z) \to \Cc = D(\Z)$ be the forgetful functor. This preserves compact objects, but it is not symmetric monoidal: we have $\Z/2Z \tensor^L\_{\Z/2\Z} \Z/2\Z \simeq \Z/2\Z$, but $\Z/2\Z \tensor^L\_{\Z} \Z/2\Z$ has higher Tor-groups so cannot be $\Z/2$.
* If $\Cc$ is cartesian closed and pointed, and $\Dd = 1$ is the trivial category, then the functor $f^\*: 1 \to \Cc: \* \mapsto \*$ is symmetric monoidal, has both adjoints $f\_!$ and $f\_\*$, and any object $C$ satisfies $f\_!(C) \simeq f\_\*(1\_{\Cc})$. But $\Cc$ cannot be tensor-triangulated (as $- \times -$ doesn't preserve coproducts in either variable.)



Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$
Consider the projective line $\mathbb{P}^1\_k$ over a field $k$. The structure morphism $f:\mathbb{P}^1\_k\rightarrow \mathrm{Spec}(k)$ induces a functor $f^\*:D(k) \rightarrow D\_{qc}(\mathbb{P}^1\_k)$ which is fully faithful (equivalently, $f\_\*(1) \simeq 1$) and which satisfies Grothendieck--Neeman duality. Since $f\_\*(1)\simeq 1$ is (trivially) a rigid separable commutative algebra, there is a canonical map $\theta:1 \to \omega\_f$ such that $f\_\*(\theta)$ an isomorphism (by Lemma 4.5 of my paper "[A characterization of finite étale morphisms in tensor triangular geometry](https://arxiv.org/abs/2106.14066)"). Thus, $f\_\*(1) \simeq f\_\*(\omega\_f) \simeq f\_!(1)$ where the last isomorphism is the Wirthmüller isomorphism provided by [BDS]. However, $\omega\_f = \Sigma \mathcal O\_{\mathbb{P}^1\_k}(-2)$ is not isomorphic to $1=\mathcal O\_{\mathbb{P}^1\_k}$.


Thus $C=\omega\_f^{-1}$ and $C=1$ both satisfy $f\_!(C)\simeq f\_\*(1)$ showing that $C$ need not be unique.


