In appendix B of this paper https://arxiv.org/abs/math/0211216, Hopkins and Singer defined the Anderson dual $\tilde{I}(E)$ of a spectrum $E$ as the function spectrum of maps from $E$ to $\tilde{I}$, where $\tilde{I}$ is a spectrum called the Anderson dual of the sphere. Then they claimed that $$[X, \tilde{I}(E)]=[X\wedge E, \tilde{I}].$$ And when we take $E$ to be the rational Eilenberg-Maclane spectrum $H\mathbb{Q}$, we have $\tilde{I}(E)=H\mathbb{Q}$.
To my understanding, this is something like $[A,[B, C]]=[A\wedge B, C]$, however, in the definition of function spectra I found on page 50 of this book https://www.math.uni-bonn.de/people/schwede/SymSpec-v3.pdf, $\tilde{I}(X)$ should be defined to be $\mathrm{map} (E, \mathrm{sh}^n(\tilde{I}))$ on the $n$-th level, where $\mathrm{sh}$ shifts the level of spectra by one. It seems there is no "homotopy class" taken in the definition of function spectrum. But in some other contexts, some authors just define this function spectrum to be $[E, \tilde{I}]$. I am quite confused about if these definitions coincide and how to deduce the statements made by Hopkins and Singer.
Any comments or corrections are welcome!