Consider a group $G$ and a ring $R$. Write $R[G]\overset{\varepsilon}{\longrightarrow}R$ for the counit map. Write $(-)_G\dashv \varepsilon ^\ast \dashv (-)^G$ for the adjoint triple on modules induced by the counit. Finally, denote the bimodule structures on $R$ induced by the counit as follows (following this answer). $$\varepsilon ^\ast R= \!\!\;_{R[G]}R_R,\;\;\varepsilon _!R[G]=\!\!\;_RR_{R[G]}$$
I would like to see that the canonical natural trasnformation $(-)^G\Rightarrow (-)_G$ (seen e.g as a canonical arrow from the apex of a limit cone to the apex of a colimit cone of a functor) is an isomorphism iff $|G|\in R^\times $.
I started with the following calculation.
$$\begin{aligned}\mathrm{Nat}((-)_{G},(-)^{G}) & \cong\mathrm{Nat}(1,\varepsilon^{\ast}\circ(-)^{G}) & (-)_{G}\dashv\varepsilon^{\ast}\\ & \cong\mathrm{Nat}(1,\negmedspace\;_{R}\mathsf{Mod}(\varepsilon_{!}R[G],(-)^{G})) & \text{def. of }\varepsilon^{\ast}\\ & \cong\mathrm{Nat}(1,\negmedspace\;_{R[G]}\mathsf{Mod}(\varepsilon^{\ast}\varepsilon_{!}R[G],-)) & \varepsilon^{\ast}\dashv(-)^{G}\\ & \cong\mathrm{Nat}(\negmedspace\;_{R[G]}\mathsf{Mod}(R[G],-),\negmedspace\;_{R[G]}\mathsf{Mod}(\varepsilon^{\ast}\varepsilon_{!}R[G],-))\\ & \cong\!\negmedspace\;_{R[G]}\mathsf{Mod}(\varepsilon^{\ast}\varepsilon_{!}R[G],R[G]) & \substack{\text{enriched}\\ \text{Yoneda} } \\ & \cong\!\negmedspace\;_{R}\mathsf{Mod}(\varepsilon_{!}R[G],R[G]^{G}) & \varepsilon^{\ast}\dashv(-)^{G}\\ & \cong\varepsilon^{\ast}(R[G]^{G}) & \text{def. of }\varepsilon^{\ast} \end{aligned}$$
This is already strange to me: the result is an $R[G]$-module, while I expected an abelian group.
At any rate, at the set level natural transformations $(-)_G\Rightarrow (-)^G$ are in bijection with $\varepsilon ^\ast (R[G]^G)$ which is itself in bijection with the $R$-module $R[G]^G$ readily seen to be generated by $\sum_{g\in G}g$ when $G$ is finite.
Questions.
- How to see the natural transformation corresponding to e.g $r\otimes s\sum_{g\in G}g\in \varepsilon ^\ast (R[G]^G)$ is given by multiplication by $sr\sum_{g\in G}g$?
- How to see the canonical natural trasnformation $(-)^G\Rightarrow (-)_G$ is an isomorphism iff $|G|\in R^\times $ with inverse given by multiplication by $\frac 1{|G|}\sum_{g\in G}g$?
