Under the analogy that $\{ \mathsf{t},\mathsf{f}\} $ should be the $\webleft (-1\webright )$-categorical analogue of $\mathsf{Sets}$, we may view a function
\[ f\colon X\to \{ \mathsf{t},\mathsf{f}\} \]
as a decategorification of presheaves and copresheaves
\begin{gather*} \mathcal{F} \colon \mathcal{C}^{\mathsf{op}} \to \mathsf{Sets},\\ F \colon \mathcal{C} \to \mathsf{Sets}.\end{gather*}
The characteristic functions $\chi _{U}$ of the subsets of $X$ are then the primordial examples of such functions (and, in fact, all of them).
