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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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