# The Stacks Project

## Tag 001O

Example 4.3.4. Functor of points. For any $U\in \mathop{\rm Ob}\nolimits(\mathcal{C})$ there is a contravariant functor $$\begin{matrix} h_U & : & \mathcal{C} & \longrightarrow & \textit{Sets} \\ & & X & \longmapsto & \mathop{\rm Mor}\nolimits_\mathcal{C}(X, U) \end{matrix}$$ which takes an object $X$ to the set $\mathop{\rm Mor}\nolimits_\mathcal{C}(X, U)$. In other words $h_U$ is a presheaf. Given a morphism $f : X\to Y$ the corresponding map $h_U(f) : \mathop{\rm Mor}\nolimits_\mathcal{C}(Y, U)\to \mathop{\rm Mor}\nolimits_\mathcal{C}(X, U)$ 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 $U$.

The code snippet corresponding to this tag is a part of the file categories.tex and is located in lines 465–495 (see updates for more information).

\begin{example}
\label{example-hom-functor}
Functor of points.
For any $U\in \Ob(\mathcal{C})$ there is a contravariant
functor
$$\begin{matrix} h_U & : & \mathcal{C} & \longrightarrow & \textit{Sets} \\ & & X & \longmapsto & \Mor_\mathcal{C}(X, U) \end{matrix}$$
which takes an object $X$ to the set
$\Mor_\mathcal{C}(X, U)$. In other words $h_U$ is a presheaf.
Given a morphism $f : X\to Y$ the corresponding map
$h_U(f) : \Mor_\mathcal{C}(Y, U)\to \Mor_\mathcal{C}(X, U)$
takes $\phi$ to $\phi\circ f$. We will always denote
this presheaf $h_U : \mathcal{C}^{opp} \to \textit{Sets}$.
It is called the {\it representable presheaf} associated to $U$.
If $\mathcal{C}$ is the category of schemes this functor is
sometimes referred to as the
\emph{functor of points} of $U$.
\end{example}

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