TITLE: What is the easiest way to see $\langle \Sigma X, Y \rangle\cong \langle X,\Omega Y\rangle $
QUESTION [5 upvotes]: Let $X$ and $Y$ be topological spaces. Let $\langle X,Y\rangle$ denote the homotopy classes of maps from $X$ and $Y$. The reduced suspension $\Sigma(-)$ has the adjoint $\Omega(-)$. In other words, we have 
$$
\langle \Sigma X, Y \rangle\cong \langle X,\Omega Y\rangle
$$
for all $X$ and $Y$. 
I am always confused with on which side I should put $\Sigma$. What is the easiest or intuitive way to think of this isomorphism? Are there a good way to memorize this this formula?

REPLY [5 votes]: The easiest way is to think about the adjunctions you do know. I like to think about the Hom-Tensor adjunction. That is all the above is (in the category of pointed spaces). Let me know if you need me to elaborate.