Let $f\colon S^n \rightarrow S^n$ be a basepoint preserving map of degree $d \geq 1$. We then get an induced map $\Omega \Sigma(f)\colon \Omega \Sigma S^n \rightarrow \Omega \Sigma S^n$. There is another map $g\colon \Omega \Sigma S^n \rightarrow \Omega \Sigma S^n$ taking a based loop $\gamma$ in $\Sigma S^n$ to $\gamma^d$, where $\gamma^d$ uses the loop product (and thus goes around $\gamma$ a total of $d$ times).

These maps seem pretty similar. Question: is $\Omega \Sigma(f)$ homotopic to $g$?

Let $f\colon S^n \rightarrow S^n$ be a basepoint preserving map of degree $d \geq 1$. We then get an induced map $\Omega \Sigma(f)\colon \Omega \Sigma S^n \rightarrow \Omega \Sigma S^n$. There is another map $g\colon \Omega \Sigma S^n \rightarrow \Omega \Sigma S^n$ taking a based loop $\gamma$ in $\Sigma S^n$ to $\gamma^d$, where $\gamma^d$ uses the loop product (and thus goes around $\gamma$ a total of $d$ times).

These maps seem pretty similar. Question: is $\Omega \Sigma(f)$ homotopic to $g$? Or at least is the diagram

$$\require{AMScd}\begin{CD} S^n @>{f}>> S^n \\ @VVV @VVV \\ \Omega \Sigma S^n @>{g}>> \Omega \Sigma S^n \end{CD}$$

commutative up to homotopy?