In this question about Hopf Invariant I was able to prove that if $g: \mathbb{S}^n \longmapsto \mathbb{S}^n$ is a map of degree $m$, then the Hopf invariant $H(g \circ f) = m^2H(f)$ using the fact that there is a natural map from $\psi : C_f \longmapsto C_{g\circ f}$ defined by $\begin{cases}(x,t) \to (x,t) \\ y \to g(y) \end{cases}$ which commutes with the squares (that are explicit in the linked question).
Now I have to prove that if $g: \mathbb{S}^{2n-1} \longmapsto \mathbb{S}^{2n-1}$ then $H(f \circ g)=mH(f)$. In order to do that I'd like to costruct an explicit map from $C_{f \circ g}\longmapsto C_f$ such the diagrams commute to induce the diagram in cohomology, but I didn't find one.
Any help or hint would be appreciated.
The setup, as I understand it, is that you're given $X \xrightarrow{g} Y \xrightarrow{f} Z$. Then define $C_{f \circ g} \to C_f$ by $z \mapsto z$ and $(x,t) \mapsto (g(x), t)$. In terms of diagrams, in case it's helpful: $$ \require{AMScd} \begin{CD} X @>{f \circ g}>> Z @>>> C_{f \circ g} \\ @VV{g}V @| @VVV \\ Y @>{f}>> Z @>>> C_f \end{CD} $$ The left square clearly commutes, and so there should be an induced map on the cofibers.