Definition 26.17.1. Given morphisms of schemes $f : X \to S$ and $g : Y \to S$ the fibre product is a scheme $X \times _ S Y$ together with projection morphisms $p : X \times _ S Y \to X$ and $q : X \times _ S Y \to Y$ sitting into the following commutative diagram

$\xymatrix{ X \times _ S Y \ar[r]_ q \ar[d]_ p & Y \ar[d]^ g \\ X \ar[r]^ f & S }$

which is universal among all diagrams of this sort, see Categories, Definition 4.6.1.

There are also:

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