I have a question regarding the Hurewicz theorem in the particular case of the spheres. The theorem is the following:
Hurewicz Theorem : Let $X$ be a topological space. There is a funtorial homomorphism $h_n : \pi_n(X,x_0) \longmapsto H_n(X)$. If $X$ is $(n-1)-$connected $h_n$ is an isomorphism, or in the case $n=1$ is the quotient by the subgroup of commutators.
In my notes $h_n$ is supposed to be known but we explicitly write it down just in the case $n=1$.I'm looking for a fast proof at least for spheres.
I was wondering whether the following argument could be use to prove that $h_n$ ($n \geq 2$) is an homomorphism in this special case. I'm going to name $\Sigma_i : \pi_i(\mathbb{S}^i) \longmapsto \pi_{i+1}(\mathbb{S}^{i+1})$ the isomorphism given by Freudenthal theorem and $(\Sigma f)_*^{-1},$ the inverse of the isomorphism given by the Mayer Vietoris exact sequence considering the suspension space, which is the case of the $n-$sphere, results to be the $(n+1)-$sphere.
$$\require{AMScd} \begin{CD} \pi_1(\mathbb{S}^1) @>\Sigma_1>> \pi_2(\mathbb{S}^2) @>\Sigma_2>> \pi_3(\mathbb{S}^3) @>\Sigma_3>> \pi_4(\mathbb{S}^4)@>\Sigma_4>> \pi_5(\mathbb{S}^5) @> >> \cdots \\ @Vh_1VV @Vh_2VV @Vh_3VV @Vh_4VV @Vh_5VV \\ H_1(\mathbb{S}^1) @>(\Sigma f)_*^{-1}>> H_2(\mathbb{S}^2) @>(\Sigma f)_*^{-1}>> H_3(\mathbb{S}^3) @>(\Sigma f)_*^{-1}>> H_4(\mathbb{S}^4)@>(\Sigma f)_*^{-1}>> H_5(\mathbb{S}^5) @> >> \cdots \end{CD} $$
Since $\Sigma_\bullet$ and $(\Sigma f)_*$ induce isomorphism for every $n$, I could start, since $h_1$ is known, defining $h_2$ to be $(\Sigma f)_*^{-1} \circ h_1 \circ \Sigma_1^{-1}$.
This definition has the advantages to let $h_2$ be an isomorphism, since the three defining are(due to $\pi_1({\mathbb{S}^1})$ abelian) and to make the diagram commute.
I doubt this works, since I know that the Hurewicz homomorphism has a well-defined definition, so this could not coincide with it, hence not good in order to continue the theorem. Plus, I don't know whether the isomorphism costructe in this way is funtorial, it should be if all the three maps are, but I'm not if $\Sigma_1$ is.
Where my construction fails? Any help or tip would be appreciated.