This is an assemblage of known results, I'll try to put a reference for all of them.

• By a classical theorem of Serre all stable homotopy groups are finite in positive degree. In particular we have $\mathbb{S}_\mathbb{Q}=H\mathbb{Q}$.

• Rationalization is a smashing localization, so that $\pi_*(\mathbb{S}_\mathbb{Q}\otimes E)\cong \pi_*E \otimes_{\mathbb{Z}}\mathbb{Q}$. This follows immediately from the fact that we can write $$E_\mathbb{Q}=\mathrm{colim}\left(E\xrightarrow{2}E\xrightarrow{3}E\xrightarrow{4}E\xrightarrow{5}\cdots\right)$$ and that tensor (smash) products and homotopy groups commute with filtered colimits of spectra.

• By Schwede-Shipley Morita theory, the category of modules over $H\mathbb{Q}$ is equivalent to the derived category of $\mathbb{Q}$. (Theorem 5.1.6 in Stable model categories are categories of modules) Moreover under this equivalence homotopy groups correspond to homology groups (because $H\mathbb{Q}$ is sent to $\mathbb{Q}$ in degree 0).

• In the derived category $D(\mathbb{Q})$ for every object $M$ there is an equivalence $M\cong \bigoplus_{n\in\mathbb{Z}} (H_nM)[n]$. This is an easy exercise, using the fact that in $\mathbb{Q}$-vector spaces every short exact sequence splits.

In particular for every spectrum $E$, the spectrum $E\otimes H\mathbb{Q}$ has homotopy groups $\pi_*(E\otimes H\mathbb{Q})\cong \pi_*E\otimes_{\mathbb{Z}}\mathbb{Q}$ and so there is an equivalence of $H\mathbb{Q}$-modules (in particular of spectra)

$$ E\otimes H\mathbb{Q}\cong \bigoplus_{n\in\mathbb{Z}} \Sigma^nH(\pi_nE\otimes_{\mathbb{Z}}\mathbb{Q})\cong\prod_{n\in\mathbb{Z}} \Sigma^nH(\pi_nE\otimes_{\mathbb{Z}}\mathbb{Q})$$ (it's not necessary, but it is very convenient to observe that in this case for degree reasons the coproduct and the product coincide)

The particular formula you are referring to follows from the rational homotopy groups of $\mathrm{bstring}$ and the fact that, for $V,W$ rational vector spaces $$[V[n],W[m]]=\begin{cases}\mathrm{Hom}_\mathbb{Q}(V,W) & \textrm{ if }n=m\\ 0&\textrm{ otherwise}\end{cases}\,.$$

This is an assemblage of known results, I'll try to put a reference for all of them.

• By a classical theorem of Serre all stable homotopy groups are finite in positive degree. In particular we have $\mathbb{S}_\mathbb{Q}=H\mathbb{Q}$.

• Rationalization is a smashing localization, so that $\pi_*(\mathbb{S}_\mathbb{Q}\otimes E)\cong \pi_*E \otimes_{\mathbb{Z}}\mathbb{Q}$. This follows immediately from the fact that we can write $$E_\mathbb{Q}=\mathrm{colim}\left(E\xrightarrow{2}E\xrightarrow{3}E\xrightarrow{4}E\xrightarrow{5}\cdots\right)$$ and that tensor (smash) products and homotopy groups commute with filtered colimits of spectra.

• By Schwede-Shipley Morita theory, the category of modules over $H\mathbb{Q}$ is equivalent to the derived category of $\mathbb{Q}$. (Theorem 5.1.6 in Stable model categories are categories of modules) Moreover under this equivalence homotopy groups correspond to homology groups (because $H\mathbb{Q}$ is sent to $\mathbb{Q}$ in degree 0).

• In the derived category $D(\mathbb{Q})$ for every object $M$ there is an equivalence $M\cong \bigoplus_{n\in\mathbb{Z}} (H_nM)[n]$. This is an easy exercise, using the fact that in $\mathbb{Q}$-vector spaces every short exact sequence splits.

In particular for every spectrum $E$, the spectrum $E\otimes H\mathbb{Q}$ has homotopy groups $\pi_*(E\otimes H\mathbb{Q})\cong \pi_*E\otimes_{\mathbb{Z}}\mathbb{Q}$ and so there is an equivalence of $H\mathbb{Q}$-modules (in particular of spectra)

$$ E\otimes H\mathbb{Q}\cong \bigoplus_{n\in\mathbb{Z}} \Sigma^nH(\pi_nE\otimes_{\mathbb{Z}}\mathbb{Q})\cong\prod_{n\in\mathbb{Z}} \Sigma^nH(\pi_nE\otimes_{\mathbb{Z}}\mathbb{Q})$$ (it's not necessary, but it is very convenient to observe that in this case for degree reasons the coproduct and the product coincide)

So, to conclude, we have that $$[bstring, X\otimes H\mathbb{Q}]\cong \left[ bstring, \prod_{n\in\mathbb{Z}}\Sigma^nH(\pi_nX\otimes_{\mathbb{Z}}\mathbb{Q})\right]\cong \prod_{n\in\mathbb{Z}} H^n(bstring;\pi_nX\otimes_{\mathbb{Z}}\mathbb{Q})\cong \prod_{n\in\mathbb{Z}} \pi_nX\otimes_{\mathbb{Z}} H^n(bstring;\mathbb{Q})\,.$$

This is an assemblage of known results, I'll try to put a reference for all of them.

• By a classical theorem of Serre all stable homotopy groups are finite in positive degree. In particular we have $\mathbb{S}_\mathbb{Q}=H\mathbb{Q}$.

• Rationalization is a smashing localization, so that $\pi_*(\mathbb{S}_\mathbb{Q}\otimes E)\cong \pi_*E \otimes_{\mathbb{Z}}\mathbb{Q}$. This follows immediately from the fact that we can write $$E_\mathbb{Q}=\mathrm{colim}\left(E\xrightarrow{2}E\xrightarrow{3}E\xrightarrow{4}E\xrightarrow{5}\cdots\right)$$ and that tensor (smash) products and homotopy groups commute with filtered colimits of spectra.

• By Schwede-Shipley Morita theory, the category of modules over $H\mathbb{Q}$ is equivalent to the derived category of $\mathbb{Q}$. (Theorem 5.1.6 in Stable model categories are categories of modules) Moreover under this equivalence homotopy groups correspond to homology groups (because $H\mathbb{Q}$ is sent to $\mathbb{Q}$ in degree 0).

• In the derived category $D(\mathbb{Q})$ for every object $M$ there is an equivalence $M\cong \bigoplus_{n\in\mathbb{Z}} (H_nM)[n]$. This is an easy exercise, using the fact that in $\mathbb{Q}$-vector spaces every short exact sequence splits.

In particular for every spectrum $E$, the spectrum $E\otimes H\mathbb{Q}$ has homotopy groups $\pi_*(E\otimes H\mathbb{Q})\cong \pi_*E\otimes_{\mathbb{Z}}\mathbb{Q}$ and so there is an equivalence of $H\mathbb{Q}$-modules (in particular of spectra)

$$ E\otimes H\mathbb{Q}\cong \bigoplus_{n\in\mathbb{Z}} \Sigma^nH(\pi_nE\otimes_{\mathbb{Z}}\mathbb{Q})$$

The particular formula you are referring to follows from the rational homotopy groups of $\mathrm{bstring}$ and the fact that, for $V,W$ rational vector spaces $$[V[n],W[m]]=\begin{cases}\mathrm{Hom}_\mathbb{Q}(V,W) & \textrm{ if }n=m\\ 0&\textrm{ otherwise}\end{cases}\,.$$

This is an assemblage of known results, I'll try to put a reference for all of them.

• By a classical theorem of Serre all stable homotopy groups are finite in positive degree. In particular we have $\mathbb{S}_\mathbb{Q}=H\mathbb{Q}$.

• Rationalization is a smashing localization, so that $\pi_*(\mathbb{S}_\mathbb{Q}\otimes E)\cong \pi_*E \otimes_{\mathbb{Z}}\mathbb{Q}$. This follows immediately from the fact that we can write $$E_\mathbb{Q}=\mathrm{colim}\left(E\xrightarrow{2}E\xrightarrow{3}E\xrightarrow{4}E\xrightarrow{5}\cdots\right)$$ and that tensor (smash) products and homotopy groups commute with filtered colimits of spectra.

• By Schwede-Shipley Morita theory, the category of modules over $H\mathbb{Q}$ is equivalent to the derived category of $\mathbb{Q}$. (Theorem 5.1.6 in Stable model categories are categories of modules) Moreover under this equivalence homotopy groups correspond to homology groups (because $H\mathbb{Q}$ is sent to $\mathbb{Q}$ in degree 0).

• In the derived category $D(\mathbb{Q})$ for every object $M$ there is an equivalence $M\cong \bigoplus_{n\in\mathbb{Z}} (H_nM)[n]$. This is an easy exercise, using the fact that in $\mathbb{Q}$-vector spaces every short exact sequence splits.

In particular for every spectrum $E$, the spectrum $E\otimes H\mathbb{Q}$ has homotopy groups $\pi_*(E\otimes H\mathbb{Q})\cong \pi_*E\otimes_{\mathbb{Z}}\mathbb{Q}$ and so there is an equivalence of $H\mathbb{Q}$-modules (in particular of spectra)

$$ E\otimes H\mathbb{Q}\cong \bigoplus_{n\in\mathbb{Z}} \Sigma^nH(\pi_nE\otimes_{\mathbb{Z}}\mathbb{Q})\cong\prod_{n\in\mathbb{Z}} \Sigma^nH(\pi_nE\otimes_{\mathbb{Z}}\mathbb{Q})$$ (it's not necessary, but it is very convenient to observe that in this case for degree reasons the coproduct and the product coincide)

The particular formula you are referring to follows from the rational homotopy groups of $\mathrm{bstring}$ and the fact that, for $V,W$ rational vector spaces $$[V[n],W[m]]=\begin{cases}\mathrm{Hom}_\mathbb{Q}(V,W) & \textrm{ if }n=m\\ 0&\textrm{ otherwise}\end{cases}\,.$$