In category of sets, an exp object { Z^{Y}} is set of all functions { Y->Z}. map { {eval} ,(Z^{Y}* Y)\to Z} is just evaluation map, which sends pair { (f,y)} to {f(y)}. For any map { g, (X* Y)-& Z} map {\lambda g, X->Z^{Y}} is #curried form of { g}:
{ \lambda g(x)(y)=g(x,y)}