Consider the site of profinite sets $\mathcal{S}$. In condensed math we consider sheaves of abelian groups $C = \text{Sh}(\mathcal{S}, \text{Ab})$ on this site. It has a natural tensor product and internal hom $[-, -]_C$ which makes it monoidal closed. We also consider the derived category of this category, call it $D$, wich has a derived tensor product and derived internal hom $[-, -]_D$.

I am thinking about objects $M$ in $C$ such that $[[M, \mathbb{R}]_C, \mathbb{R}]_C \cong M$, and objects $M$ in $D$ such that $[[M, \mathbb{R}]_D, \mathbb{R}]_D \cong M$. Call these reflexive.

I am also interested in the property of $[M \otimes N, \mathbb{R}]_C \cong [M, \mathbb{R}]_C \otimes [N, \mathbb{R}]_C$ and $[M \otimes N, \mathbb{R}]_D \cong [M , \mathbb{R}]_D \otimes [N, \mathbb{R}]_D$.

Using these properties and not much extra work, one can show algebraically nice versions of fourier duality and several other important theorems in analysis. Considering algebras and coalgebras in this category (and bialgebras and hopf algebras) gives ways of showing pontryjagin dualilty and the existence of haar measure in a pretty sleek way, at least for those objects which satisfy the two properties above.

I think it should be true of a broad class of objects $M$ in $D$ that $[M, \mathbb{R}]_D$ is reflexive. When restricted to these objects, $[-, \mathbb{R}]_D$ would form an idempotent adjunction. Maybe someone can hep me to show this in the simplest way - perhaps by showing that solidification is idempotent and matches double dual on this class. Maybe there is also a version of this in $C$?

The details are a bit hard to nail down here, at least for me. It'd be nice to see how this works in detail for the free object on a profinite set $S$, too. I think this doube dualizes the profinite limit of $\mathbb{R}$'s over $S$, which somehow is reflexive?

Can anyone give an outline of how one might go about these things?