The Stacks project

Example 59.9.2. Given an object $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, we consider the rule

\[ \begin{matrix} U & \longmapsto & h_ X(U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, X) \\ V \xrightarrow {\varphi } U & \longmapsto & (\psi \mapsto \psi \circ \varphi ) : h_ X(U) \to h_ X(V). \end{matrix} \]

This defines a functor $h_ X : \mathcal{C}^{opp} \to \textit{Sets}$ and hence a presheaf. This is called the representable presheaf associated to $X$. It is not true that representable presheaves are sheaves in every topology on every site.

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