Given two spectra $A$ and $B$, the set $[A,B]$ of homotopy classes of maps from $A$ to $B$ forms an abelian group. Can the dual abelian group $\text{Hom}([A,B],\mathbb{Q}/\mathbb{Z})$ be expressed as a group of homotopy classes of maps between spectra?

The *Brown-Comenetz dual* $I_{\mathbb{Q}/\mathbb{Z}}E$ of a spectrum $E$ is defined here. A key property of $I_{\mathbb{Q}/\mathbb{Z}}E$ is that its homotopy is dual of that of $E$ in the sense that $\pi_*I_{\mathbb{Q}/\mathbb{Z}}E=\text{Hom}(\pi_*E,\mathbb{Q}/\mathbb{Z})$.

Naively, I suspect that $\text{Hom}([A,B],\mathbb{Q}/\mathbb{Z})$ can be expressed in terms of $I_{\mathbb{Q}/\mathbb{Z}}A$ and $I_{\mathbb{Q}/\mathbb{Z}}B$.

What if I replace ${\mathbb{Q}/\mathbb{Z}}$ by ${\mathbb{R}/\mathbb{Z}}$ or ${\mathbb{C}/\mathbb{Z}}$?