I am studying products in the category of affine varieties and I don't know how to prove that the map
$ \mathbb{K}[X] \otimes_{\mathbb{K}} \mathbb{K}[Y] \longrightarrow \mathbb{K}[X \times Y], $
where $X$ is a affine variety and $ \mathbb{K}[X]$ is the ring of regular function on $X$, is surjective.
I know that if $W \subset \mathbb{A}^n$ is a closed subset then $ \mathbb{K}[W] \simeq \mathbb{K}[z_1, \dots,z_n]/I(W)$ but I can't link the two results.
Have you any hints?