Point-lifting property of a scheme fibre product ($\exists z\mapsto (x,y)⟺f(x)=g(y)$)

For morphisms of schemes $X\to S$, $Y\to S$, a pair of points $x,y$ lifts to a point of $X{\times }_{S}Y$ 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.