I am trying to show that the projections $\mathbb{A}^2\to\mathbb{A}^1$ exhibit $\mathbb{A}^2$ as $\mathbb{A}^1\times\mathbb{A}^1$ in the category of affine algebraic sets, for which I have to first show that the aforementioned projections are morphisms of affine algebraic sets.
I know the definition of morphism, which is: given affine algebraic sets $V\subset\mathbb{A}^n$, $W\subset\mathbb{A}^m$ over an algebraically closed field $K$, a map $f:V\to W$ is a morphism if there are polyomials $p_i\in K[X_1,X_2,\cdots,X_n]$, $i\in\{1,2,\cdots,m\}$ such that for every $v\in V$, $f(v)=(p_1(v),p_2(v),\cdots,p_m(v))$.
First, of all do I need to use the explicit projections to show that the projections $\mathbb{A}^2\to\mathbb{A}^1$ exhibit $\mathbb{A}^2$ as $\mathbb{A}^1\times\mathbb{A}^1$? Or any, abstract argument is enough? If any abstract argument is enough, what do I argue here? If not, then what are the required projections and the polynomials?
$\def\A{\mathbb{A}}$ We work over the field $K$. Let us consider the projection map $f:(x,y)\in\A^2\mapsto x\in A^1$ and the polynomial $p(X,Y)=X\in K[X,Y]$. It should not be difficult to show that $f(v)=(p(v))$ for all $v\in\A^2$ (on the right of the equal sign I wrote a «$1$-uple» $(p(v))$ so as to match with what you wrote in the question as definition of morphism, but certainly it is just notation).
Can you show that the map $g:(x,y)\in\A^2\mapsto(x^2+y,xy^2,3-x^2y)\in\A^3$ is a morphism of varities? What are the polynomials that you need to do that?