= Solution
Choose projective embeddings $X\subseteq\mathbb P^n$ and $Y\subseteq\mathbb P^m$. Define their <product> as the Segre <image> of the <set> of pairs $(x,y)$, with its induced <variety> structure.
It is closed in the Segre <variety>. To see this directly, for every homogeneous $f\in I(X)$ impose the equations $f(z_{0j},\ldots,z_{nj})=0$ on every column, and for every homogeneous $g\in I(Y)$ impose $g(z_{i0},\ldots,z_{im})=0$ on every row. For a <rank-one matrix> $z_{ij}=x_iy_j$, the column equations are $y_j^{\deg f}f(x)=0$. A nonzero $y_j$ therefore forces $x\in X$. The row equations similarly force $y\in Y$. Thus these homogeneous equations together with the <minors> cut out exactly $X\times Y$, proving it is projective.
On a chart where $z_{i_0j_0}\neq0$, the projections have formulas
$$
\pi_X([Z])=[z_{0j_0}:\cdots:z_{nj_0}],\qquad
\pi_Y([Z])=[z_{i_00}:\cdots:z_{i_0m}].
$$
These <homogeneous coordinate> tuples are nonzero on the chart and agree on overlaps, so both projections are <morphism of varieties>[morphisms].
Given <morphism of varieties>[morphisms] $\alpha:Z\to X$ and $\beta:Z\to Y$, define $\gamma(z)=\sigma_{n,m}(\alpha(z),\beta(z))$. Locally write their coordinates as $[a_0:\cdots:a_n]$ and $[b_0:\cdots:b_m]$. Then $\gamma$ has coordinates $[a_ib_j]$, which are regular with a nonzero coordinate near each <point>. Hence $\gamma$ is a <morphism of varieties>[morphism] and
$$
\boxed{\pi_X\circ\gamma=\alpha,\qquad\pi_Y\circ\gamma=\beta.}
$$
The two projections determine the underlying pair uniquely, so this <morphism of varieties>[morphism] is unique. This is the <product> <universal property>, and it makes the construction independent up to canonical <isomorphism> of the chosen embeddings.
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