Higher homotopy groups are iterated loop components

Iterate the interval transpose of F1 n times. A boundary-constant $a:I^n\to X$ becomes a point of ${\mathrm{\Omega }}^{n}X$: fixing any interval endpoint makes the appropriate loop constant. Conversely evaluation at all n parameters recovers a. The inverse identities follow one coordinate at a time. F2 identifies these with the stated spherical homotopy groups as well.

Apply the same transpositions with a further time variable. Boundary-fixed cubical homotopies become paths in ${\mathrm{\Omega }}^{n}X$, and paths evaluate to such homotopies. Thus the map induces inverse bijections between cube classes and components. On the two halves of coordinate 1 the evaluated concatenation is exactly $a(2s_1,u)$ or $b(2s_1-1,u)$, so the component operation matches the cubical group operation of F3. Evaluation also commutes with based postcomposition, giving naturality.