The evaluation map

Let S be a fixed set. For each set X, let XS denote the set of all functions h:SX in Set.

  1. Show that XXS is the object function of a functor SetSet.
  2. Show that XXS×S is the object function of a functor SetSet.
  3. For each set X let eX:XS×SX be the evaluation map, defined by e(h,s)=h(s). Show that these maps are the components of a natural transformation e:S×SI, where I is the identity functor on Set.