So when defining a Cohomology theory of a spectrum you define the addition structure via the pinch map. I.E. to define addition on $[X,E_n]$ look at $f,g \in [\sum X, E_{n+1}]$ let $\iota: \sum X \to \sum X \vee \sum X$ then $f+g=\iota^*(f\vee g)$. Identity is given by the constant map and inverses are apparently given by reversing the direction of $S^1$ in $\sum X=S^1 \wedge X$: $-f(s,x)=f(-s,x)$.
What is the homotopy between the $f-f$ and the identity a constant map? I would imagine this is the same for the higher homotopies.
Should I use adjointness? I would rather have a direct homotopy.