Let $k$ be an algebraically closed field.

(i) Let $X$ and $Y$ be affine varieties defined over $k$. Given a map $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of affine varieties.

(ii) With $X, Y$ still affine varieties over $k$, show that there is a one-to-one correspondence between $\operatorname{Hom}(X, Y)$, the set of morphisms between $X$ and $Y$, and $\operatorname{Hom}(A(Y), A(X))$, the set of $k$-algebra homomorphisms between $A(Y)$ and $A(X)$.

(iii) Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{4}$ be given by $f(t, u)=\left(u, t, t^{2}, t u\right)$. Show that the image of $f$ is an affine variety $X$, and find a set of generators for $I(X)$.