where is the set of homotopy classes of maps , and is the suspension of A, and is the loop space of A. In essence, the suspension can be seen as the cartesian product of with the unit interval, modulo an equivalence relation to turn the interval into a loop. The curried form then maps the space to the space of functions from loops into , that is, from into . Then is the adjoint functor that maps suspensions to loop spaces, and uncurrying is the dual.
where is the set of homotopy classes of maps  and is the suspension of a and is the loop space of a in essence the suspension can be seen as the cartesian product of with the unit interval modulo an equivalence relation to turn the interval into a loop the curried form then maps the space to the space of functions from loops into  that is from into  then is the adjoint functor that maps suspensions to loop spaces and uncurrying is the dual