Show that $\mathbb{A}^2 \cong \mathbb{A}^1 \times \mathbb{A}^1$ as varieties. Show that $\mathbb{A}^2 \cong \mathbb{A}^1 \times \mathbb{A}^1$ as varieties.
I know how to do this in the case of affine varieties. Since $\mathbb{A}^1 \times \mathbb{A}^1$ is a product in the category of affine varieties, its coordinate ring $k[\mathbb{A}^1 \times \mathbb{A}^1]$ is isomorphic to $k[x]\otimes_k k[y]$, a co-product in the category of $k$-algebras. Since $k[\mathbb{A}^2]\cong k[x,y] \cong k[x]\otimes_k k[y]$ as $k$-algebras, we are done.
How do I generalize this to varieties? Thanks.
 A: Let me just rephrase what Hoot has already said in the comments.
Namely, suppose that $X$ is some $k$-variety, and we have morphisms $f,g:X\to \mathbb{A}^1$. Then, these correspond to $k$-algebra maps $f^\ast,g^\ast:k[x]\to\mathcal{O}_X(X)$. Thus, they factor uniquely through the map $f^\ast\otimes g^\ast:k[x,y]\to \mathcal{O}_X(X)$ which corresponds to a map $X\to\mathbb{A}^2$ giving you the result.
In slightly fancier language, notice that we have functors
$$\left(k\mathsf{-Varieties}\right)^\text{op}\overset{\displaystyle \overset{\mathrm{Spec}}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}k\mathsf{-Algebras}$$
Which are adjoints of one another:
$$\mathrm{Hom}_{(k\mathsf{-Varieties})^\text{op}}(\mathrm{Spec}(A),X)=\mathrm{Hom}_{k\mathsf{-Algebras}}(A,\Gamma(X))$$
In particular, we see that $\mathrm{Spec}$ is a left-adjoint, and so by basic category theory $\mathrm{Spec}(A\sqcup B)=\mathrm{Spec}(A)\sqcup \mathrm{Spec}(B)$ where I am using $\sqcup$ to denote the arbitrary notion of coproduct. But, what is the coproduct in $k\mathsf{-Algebras}$? It's just tensor product! So, this says that 
$$\mathrm{Spec}(A\otimes_k B)=\mathrm{Spec}(A)\sqcup\mathrm{Spec}(B)$$
where, be careful, as we've set it up, the coproduct on the right hand side is the coproduct in the opposite category of $k\mathsf{-Varieties}$! But, a coproduct in the opposite category is just a product in the original category. Thus, we see that 
$$\mathrm{Spec}(A\otimes_k B)=\mathrm{Spec}(A)\times\mathrm{Spec}(B)$$
where the right hand side is now the product in the honest category $k\mathsf{-Varieties}$—i.e. fiber product over $k$.
