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 $[M \otimes N, \mathbb{R}]_D \cong [M , \mathbb{R}]_D \otimes [N, \mathbb{R}]_D$.

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?

I think that, for a topological vector space $M$, $[M, \mathbb{R}]_{C}$ arises from the dual with the compact open topology. And if that's true, then we retain the banach space - smith space duality.

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

P.S. there's a lot of clues from the notes here. I was thinking that there could be a bit of an advantage asking here since I might get the most generic characterization of the reflexive property. It would also be nice if someone could confirm that the compact open topology matches the internal hom for condensed abelian groups (when the dualizing object is $\mathbb{R}$).

P.P.S. one thing that occurs to me is that, if the stalks are finite dimensional $\mathbb{R}$-vector spaces, then we get the desired reflexive property.