The evaluation map

Let $S$ be a fixed set. For each set $X$, let $X^S$ denote the set of all functions $h:S\to X$ in $\mathbf{S}\mathbf{e}\mathbf{t}$.

1. Show that $X\mapsto X^S$ is the object function of a functor $\mathbf{S}\mathbf{e}\mathbf{t}\to \mathbf{S}\mathbf{e}\mathbf{t}$.
2. Show that $X\mapsto X^S\times S$ is the object function of a functor $\mathbf{S}\mathbf{e}\mathbf{t}\to \mathbf{S}\mathbf{e}\mathbf{t}$.
3. For each set $X$ let $e_X:X^S\times S\to X$ be the evaluation map, defined by $e(h,s)=h(s)$. Show that these maps are the components of a natural transformation $e:{\bullet }^S\times S\Rightarrow I$, where $I$ is the identity functor on $\mathbf{S}\mathbf{e}\mathbf{t}$.