Products of classical algebraic sets and their universal property

Fix the page's algebraically closed field $k$. Let $\mathcal{C}$ be the category whose objects are classical affine or projective algebraic sets over $k$ (including empty and reducible ones), and whose arrows are regular $k$-maps. Objects isomorphic to such sets are understood with their transported algebraic structure. No existence of products for arbitrary mixed affine/projective factors is asserted.

For $X,Y$ in $\mathcal{C}$, a constructed product $X{\times }_{k}Y$ is an object of $\mathcal{C}$ with morphisms $p:X{\times }_{k}Y\to X$ and $q:X{\times }_{k}Y\to Y$ such that, for every object $T$ of $\mathcal{C}$ and morphisms $f:T\to X$, $g:T\to Y$, there is a unique morphism $⟨f,g⟩:T\to X{\times }_{k}Y$ satisfying $p⟨f,g⟩=f$ and $q⟨f,g⟩=g$. Thus it is the categorical product of Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations in $\mathcal{C}$.

The underlying set is written as pairs when a construction supplies that identification. If either factor is empty, the product set is empty. A product with the one-point affine algebraic set has the evident projection isomorphism. When $X,Y$ are varieties, this definition is used only after a construction shows that the resulting nonempty algebraic set is irreducible.