  Then this system has a solution, and we take ψ(f) = x0†. It is then possible to show that the composition in one direction, ψ  ⋅  φ ,  is the identity on N∞ and the other composition  φ ⋅  ψ is the identity on NN.  In plainer terms, we can pass from streams to functions on numbers, and we can also go the other way.