The stable homotopy groups of the spheres $\pi_{*}^{s}$ assemble into a graded ring
$\pi_{*}^{s} = \bigoplus_{n\geq 0} \pi_{n}^{s},$
with the graded product defined `in terms of composition'. For example, if $\alpha \in \pi_{i}^{s} $ has representative $[f: S^{n+i} \rightarrow S^{n}]$ and $\beta \in \pi_{j}^{s} $ has representative $[g: S^{n+j} \rightarrow S^{n}]$, then the product $\alpha \cdot \beta \in \pi_{i+j}^{s}$ may be defined as
$\alpha \cdot \beta := [g \circ \Sigma^{j}f : S^{n+i+j} \rightarrow S^{n}]$.
But we could have also defined the product as
$\alpha \cdot \beta := [f \circ \Sigma^{i}g : S^{n+i+j} \rightarrow S^{n}]$.
(The point is that we define the product in terms of a composition, possibly after suspending.)
My question is how do you show that these two definitions define the same product?
Reading the comment to this question, one may try to approach this via an Eckmann-Hilton Duality argument. But using this argument as stated, one would end up proving that these products are commutative. This is not true, as they are graded commutative.
Maybe there is a `graded' Eckmann-Hilton duality argument I could apply, but my attempts so far have lead me to conclude the above two products only coincide up to sign, which clearly should not me true. I am clearly going wrong somewhere, but I can't figure out where.
I tried looking in the literature, but I could not find anyone who comments on this, so any help would be much appreciated.