The Stacks project

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

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 $U$.