Given two spectra $X,Y\in \mathrm{HoSpectra}$ we have an abelian group $[X,Y]$. What can be said about the Eilenberg-Maclane spectrum of $[X,Y]$? I think that $H[X,Y]$ would be $F(X,Y)$ the internal hom. Does anybody know where I can find this?

@AliCaglayan this is clearly false in general. E.g. [S^0, S^1] = 0 but F(S^0, S^1) = S^1 \ne 0

@TomBachmann I am not following how this relates to spectra/

I'm writing S^n also for their corresponding infinite suspension spectra

@AliCaglayan For any spectrum X, F(S^0,X)=X where S^0 is the sphere spectrum. So if what you say were true, all spectra would be Eilenberg-MacLane spectra. This is quite clearly false

@TomBachmann @DenisNardin thank you for your help that makes sense.