Expanding upon Remark 2.5.1.1.2, we may think of the characteristic function
\[ \chi _{x}\colon X\to \{ \mathsf{t},\mathsf{f}\} \]
of an element $x$ of $X$ as a decategorification of the representable presheaf and of the representable copresheaf
\begin{align*} h_{X} & \colon \mathcal{C}^{\mathsf{op}} \to \mathsf{Sets},\\ h^{X} & \colon \mathcal{C} \to \mathsf{Sets}\end{align*}
associated of an object $X$ of a category $\mathcal{C}$.
