I asked this on mathstackexchange but didn't get any response (or many views) so I'm asking it here, although clearly it belongs over there.

In the answer to this question on mathoverflow, it says: "The integral homology group $H_i(KU)$ is the direct limit of

$$\dots \to H_{2n+i}(BU)\to H_{2n+2+i}(BU)\to\dots,"$$

My question is why? Here is my understanding: Given two spectra $E, X$, I believe the definition of the homology group $E_i(X)$ is $$E_i(X):=[\Sigma^i \mathbb{S}, E \wedge X]_{stable}=colim_j[S^{i+j}, (E \wedge X)_j]$$ where the square brackets denote homotopy classes of maps. In the case at hand, $E=H\mathbb{Z}$ (or maybe $E=\Sigma H\mathbb{Z}$?) is the spectrum whose $n$th space is the Eilenberg-MacLane space $K(\mathbb{Z}, n)$, and $X=KU$ is the spectrum whose $n$ space is either $BU \times \mathbb{Z}$ or $BU$, depending on if $n$ is even or odd, respectively. I have no idea what the $n$th space of $HZ \wedge KU$ is, since I am confused by smash product of spectra. I'm still learning how to best think of maps between spectra.