Associativity of a homology ring of an H-space. Let $X$ be a based topological space with $\mu:X\times X\rightarrow X$ which is homotopy associative and unital wrt the basepoint. We call such $(X,\mu)$ an associative H-space. The multiplication on $H_*(X)$ is defined as $H_*(X)\otimes H_*(X)\rightarrow H_*(X\times X)\rightarrow H_*(X)$ with the second map being induced by $\mu$ and the first one by the Kunneth theorem. Is there a way to prove the associativity of this multiplication without referring to the definition of a Eilenberg-Zilber map which is used in the Kunneth theorem? I'm looking for a proof which uses naturality or some other properties of a map $H_*(X)\otimes H_*(X)\rightarrow H_*(X\times X)$ if it is possible.
This question comes from section 3.C of Hatcher where he proves that homology and cohomology of an $H$-space are Hopf algebras. He proves just the counit axim for the cohomology and I was trying to fill all details for other axioms without going to the actual definition of Eilenberg-Zilber map which is rather technical.
Any references also would be helpful. 
 A: Let $X$ be a homotopy associative $H$-space with multiplication $\mu$. All homology and cohomology is taken with coefficients in a fixed field. I was unsure in the comments if it was the associativity of the homology cross product that you were unsure of, or something else. I suggested approaching the dual problem of coassociativity of the coproduct in cohomology since I know that Hatcher proves that cup products are associtive (and these are the same as the cohomology cross product). Anyway, I'll use homology in this post, and if I've misunderstood the content of your question please let me know.
I'll write $m$ for the Pontryagin product, which is the composition
$$m:H_*X\otimes H_*X\xrightarrow\times H_*(X\times X)\xrightarrow{\mu_*}H_*X$$
where $\times$ is the homology cross product and $\mu:X\times X$ is the homotopy associative H-space multiplication. Then directly from the definitions we have a commutative diagram of abelian groups and homomorphisms
$\require{AMScd}$
\begin{CD}
    H_*X\otimes H_*X\otimes H_*X@>1\otimes m>> H_*X\otimes H_*X@>m>> H_*X\\
    @V\cong V 1\otimes\times V   @V= V  V&@V= V  V\\
    H_*X\otimes H_*(X\times X) @>1\otimes\mu_*>> H_*X\otimes H_*X@>m>>H_*X\\
    @V\cong V\times V   @V\cong V\times  V&@V= V  V\\
    H_*(X\times X\times X) @>(1\times\mu)_*>> H_*(X\times X)@>\mu_*>>H_*X.
\end{CD}
The only place that commuativity is not obvious is the bottom-left square, and here it follows from the naturality of the homology cross product, a statement of which can be found as Lemma 3B.2 (pg. 270).
Now we know that homotopic maps induce the same maps on homology, and as we have assumed the existence of an associating homotopy
$$\mu\circ(1\times\mu)\simeq\mu\circ(\mu\times 1)$$
we can use  functorality of the induced homomorphisms to get
$$\mu_*(1\times\mu)_*=(\mu\circ(1\times\mu))_*=(\mu\circ(\mu\times1))_*=\mu_*(\mu\times1)_*$$
for the maps on the bottom row of our diagram.
Therefore if we return to our first diagram and change the bracketing to get a second commuative diagram
$\require{AMScd}$
\begin{CD}
    H_*X\otimes H_*X\otimes H_*X@>m\times 1>> H_*X\otimes H_*X@>m>> H_*X\\
    @V\cong V \times\otimes 1 V   @V= V  V&@V= V  V\\
    H_*(X\times X)\otimes H_*X @>\mu_*\otimes 1>> H_*X\otimes H_*X@>m>>H_*X\\
    @V\cong V\times V   @V\cong V\times  V&@V= V  V\\
    H_*(X\times X\times X) @>(\mu\times 1)_*>> H_*(X\times X)@>\mu_*>>H_*X
\end{CD}
then we find that we can paste this new diagram and our first diagram together along their bottom rows using the commutative square
$\require{AMScd}$
\begin{CD}
    H_*(X\times X\times X)@>(1\times\mu)_*>> H_*(X\times X)\\
    @VV(\mu\times_1)_* V   @VV \mu_* V\\
    H_*(X\times X) @>\mu_*>> H_*(X).
\end{CD}
At this stage we're flying so far above my AMScd skills that I'm not going to attempt any further diagrams, but hopefully I've done enough to convince you that the result is true.
