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Example 4.3.4. Functor of points. For any U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) there is a contravariant functor

\begin{matrix} h_ U & : & \mathcal{C} & \longrightarrow & \textit{Sets} \\ & & X & \longmapsto & \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U) \end{matrix}

which takes an object X to the set \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U). In other words h_ U is a presheaf. Given a morphism f : X\to Y the corresponding map h_ U(f) : \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(Y, U)\to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U) takes \phi to \phi \circ f. We will always denote this presheaf h_ U : \mathcal{C}^{opp} \to \textit{Sets}. It is called the representable presheaf associated to U. If \mathcal{C} is the category of schemes this functor is sometimes referred to as the functor of points of U.


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