I'm currently studying Ando-Hopkins-Rezk's work Multiplicative orientations of KO-theory and of the spectrum of TMFs. At a point a presumably obvious isomorphism is mentioned, which I'm however not able to immediately see (I guess I'm missing something obvious here, forgive me if the question is too naive).

I'm referring to the identification $$ s: [bstring, X\otimes \mathbb{Q}] \cong \mathbf{D}(X):=\left\{s\in\prod_{k\geq 4} \pi_{2k}X\otimes \mathbb{Q}\,\biggr\vert\, s_k=0 \text{ if $k$ is odd}\right\} $$ (equation (5.10) in the article), where $X$ is a spectrum and $bstring$ is the spectrum realizing the standard infinite loop space structure on $BString=BO\langle 8\rangle$.

While I clearly see there's a natural morphism $s: [bstring, X\otimes \mathbb{Q}] \to \mathbf{D}(X)$, that fully encodes the datum of the sequence of group homomorphisms $$ \phi_{\ast,n}: \pi_n(bstring) \to \pi_n(X)\otimes \mathbb{Q} $$ associated with (the homotopy class of) a morphism $\phi\colon bstring \to X\otimes \mathbb{Q}$, I don't clearly see why any such a sequence should be induced by a unique (homotopy class of a) morphism from $bstring$ to $X_\mathbb{Q}$, i.e., I don't immediately see why $s$ should be a bijection. I guess the answer is in some property of $bstring_\mathbb{Q}$ so well known not to be explicitly mentioned in the article, or in some even more fundamental property of rational homotopy theory of infinite loop spaces (or even general spectra) I'm missing at the moment, but I have not been able to work out this answer myself.

Thanks a lot for any possible hint to help me solve my puzzlement