In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map 
$$
\phi: Hom\_T(X,\Omega Y)\to Hom\_T(SX,Y)
$$
is defined. And a map
$$
\eta\_n=\phi^{-n}(1\_{S^nX}):X\to \Omega^n S^nX.
$$


Question: I do not quite understand what does $\eta\_n$ mean? What means $\phi^{-n}(1\_{S^nX})$?


What means $\phi^{-n}$?



iterated loop spaces and configuration spaces
$\Phi^n:Hom(X,\Omega^nS^nX)\to Hom(SX,\Omega^{n-1}S^nX)\to \dots\to Hom(S^{n-1}X,\Omega S^nX) \to Hom(S^nX,S^nX)$.


