Following our discussion of glueing schemes: Categorical descriptions for glueing sheaves and schemes. We now discuss the construction of the fibre product of schemes by glueing.

Given arbitrary schemes , let and be the given morphisms. Let be an open affine cover of . Let , , choose an affine open cover for and an affine open cover for . The fibre product is constructed by glueing various together.

We rewrite the glueing construction of fibre product in a more categorical way as follows. Note that the colimit here is glueing construction and the consequences of the two pullback squares should be clear thinking in terms of schemes.

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Theorem[Thm 3.3, [1]/ Thm 9.1.1, [2]] For any two schemes and over a scheme , the fibre product exists and is unique up to unique isomorphism.