I am reading Smith's book on Algebraic Geometry. In Chapter 2 is a proof, more of a sketch really, of the equivalence of the category of finitely generated reduced $k$-algebras and the category of affine algebraic varieties. Most of this is easy to follow, with the exception of the uniqueness of the constructed morphism $F: V \rightarrow W$ (described below), where $V \subset \mathbb{A}^{n}$ and $W \subset \mathbb{A}^{m}$ are the corresponding affine varieties to the $k$-algebras $k[V]$ and $k[W]$, and $\phi: k[W] \rightarrow k[V]$ is a $k$-algebra morphism.
$F$ must be defined so that $f \circ F = \phi(f)$ for any polynomial $f \in k[x_{1}, \ldots, x_{m}]$; i.e. so that $\phi$ is the pullback of $F$. The way Smith does this is to define $F := (F_{j})_{j=1}^{m}$, where $F_{j} := \phi(x_{j})$ for $1 \leq j \leq m$. It's easy to check that $F$ is indeed a morphism $V \rightarrow W$ such that $f \circ F = \phi(f)$.
The part that is missing in Smith's text is the uniqueness. I can suppose $G: V \rightarrow W$ is another morphism so that $f \circ G = \phi(f) = f \circ F$, but I'm lost from here. $f$ need not be injective, so there must be another way to show $F = G$.
To show $F=G$, it suffices to show each component is the same. But $$ F_j=x_j\circ F=\phi(x_j)=x_j\circ G=G_j. $$