Pontrjagin duality gives a double dual theorem for "hom with $S^1$", and $S^1$ is $\textbf{B}\mathbb{Z}$ up to homotopy.

$\textbf{B} \textbf{B}\mathbb{Z}$, modeled by $\mathbb{C}\mathbb{P}^{\infty}$, is an H-space, and an internal abelian group in the derived category. I am interested in algebras for the monad of $[-,\textbf{B} \textbf{B}\mathbb{Z}]$, and producing a double-dual theorem of a similar nature.

It could go like this: let $[-,\textbf{B} \textbf{B}\mathbb{Z}]\text{-alg}$ be the category of algebras for $[-,\textbf{B} \textbf{B}\mathbb{Z}]$. I am looking to endow this with an internal hom $[-,-]_{[-,\textbf{B}\textbf{B}\mathbb{Z}]\text{-alg}}$ and then to demonstrate a categorial equivalence like so:

$$[-,\textbf{B} \textbf{B}\mathbb{Z}]_{[-,\textbf{B}\textbf{B}\mathbb{Z}]\text{-alg}} : [-,\textbf{B}\textbf{B}\mathbb{Z}]\text{-alg} \leftrightarrow \text{H-Spc} : [-,\textbf{B}\textbf{B}\mathbb{Z}]_{\text{H-Spc}}$$

In the above, $\text{H-Spc}$ is the homotopy category of H-spaces.

Some ideas:

1. Maybe only the H-space structure of $\textbf{B} \textbf{B}\mathbb{Z}$ is necessary.
2. It could be fruitful to use the relationship between $[-,\textbf{B}\textbf{B}\mathbb{Z}]\text{-alg}$ and $K$-theory.