$\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\tensor}{\otimes}$ $\newcommand{\colim}{\rm colim}$ $\newcommand{\Sp}{Sp}$ $\newcommand{\iHom}{\underline{\rm Hom}}$ (This is the follow-up of this question. 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, 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.)

1more comment