 Suspension-loop adjunction

For based spaces there is a natural bijection $[\Sigma X,Y]\cong [X,\Omega Y]$. A map on the suspension sends a point $x$ and suspension coordinate $t$ to a loop $t\mapsto f(x,t)$; the collapsed ends and basepoint give based loops. The reverse construction evaluates such loops. The two constructions commute with based homotopies. Applied to spheres it gives ${\pi }_{n+1}Y\cong {\pi }_n\Omega Y$.