Studying homotopy groups, I faced the definition of pointed homotopy, denoting the homotopy groups as $[\mathbb{S}^n, X]^0$.
In order to define a well-defined structure of group, we can think to $\mathbb{S}^n$ as $I^n/\partial I^n$ and $[\mathbb{S}^n, X]^0 = [(I^n,\partial I^n)(X,x_0)]$ maps of pair.
Explicit homotopies can be given to prove that $e+f \sim f+e \sim f$, $f + (-f) \sim e$, $(f+g)+h \sim f+(g+h)$ and commutativity.
I wrote those homotopies by hand heavily using convexness of $I^n$. In my notes I have an observation which states that with the same process we can consider something more general, as $[\mathbb{S}^n \wedge X,Y]^0$ which of course gives the homotopy groups as a particular case taking $X = \mathbb{S}^0$. I was wondering who are those groups? There's an easy way to show the structure of groups as in the usual homotopy groups?
The set $[X, Y]$ is a group whenever $X$ is a $H$-cogroup (i.e., cogroup object in the homotopy category of pointed spaces). In particular, all suspensions are $H$-cogroups, with the comultiplication given by the "pinch" map. So $[S^n \wedge X, Y]$ is a group if $n \geq 1$.