Fiber Products and Zariski Sheaves

Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also morphisms πX : X → Z, πY : Y → Z. Given this data, we say that an object P of C, together with morphisms p1 : P → X, p2 : P → Y is a fiber product of X with Y over Z if it satisfies the following universal property: For every object T ∈ Obj(C), and every pair of morphisms f : T → X, g : T → Y such that πX ◦ f = πY ◦ g, there exists a unique morphism h : T → P such that f = p1 ◦ h and g = p2 ◦ h. In this case, we write P as X ×Z Y . Thus, a fiber product represents a functor, which we will denote by X ×Z Y , and is unique if it exists. This much is true in any category, but existence is a question with much more substance. In particular, there are categories in which fiber products do not exist. Our immediate goal is to prove: