Point-lifting property of a scheme fibre product
= Point-lifting property of a scheme fibre product
{title2=$\exists z\mapsto(x,y)\iff f(x)=g(y)$}
For <morphisms of schemes> $X\to S$, $Y\to S$, a pair of points $x,y$ lifts to a point of $X\times_SY$ exactly when their base images agree. For sufficiency use the nonempty <spectrum of a commutative ring> of $\kappa(x)\otimes_{\kappa(s)}\kappa(y)$, by <tensor product of field extensions is nonzero>. This establishes existence, not uniqueness, of a lift.