Given a map $\omega: A\to \Omega X$, one can set up the diagram

and construct the map $\sigma : \Sigma A\to X$. It's pretty easy to check that the homotopy classes of $\omega$ and $\sigma$ correspond under the isomorphism $[A, \Omega X]\cong [\Sigma A, X]$ given by the exponential law. I'd like a good reference for this fact.