(I think I'm missing something very simple). Let $R$ be a ring and $HR$ the associated Eilenberg-Maclane spectrum, defined by $$[\Sigma^\infty_+ X,HR]_{-*}={H}^*(X; R)$$ for any CW-complex $X$, and where ${H}^*(X; R)$ denotes ordinary cohomology with $R$ coefficients.
My questions is, do we have the following equality and if so, why $$[S, HR \wedge \Sigma^\infty_+ X]_* = {H}_*(X; R) \ ?$$
The left hand side is what $HR_*(\Sigma X)$ is defined to be, so that is why I suspect it should equal the right hand side, i.e ordinary homology with $R$ coefficients, but I don't see how that follows.
As Mariano suggested in the comment, the reason that the equation is true is because the functor $X \mapsto [S, HR \wedge \Sigma_+^\infty X]$ satisfies the Eilenberg-Steenrod axioms for homology, including the dimension axiom: if $X$ is a point, then $\Sigma^\infty_+ X=S$, the sphere spectrum, and hence the value of this functor is $$[S, HR \wedge S]_*=[S, HR]_*=\pi_*(HR) =\left\{ \begin{array}{ll} R & *=0 \\ 0 & \text{otherwise} \end{array} \right. $$