Expanding upon Remark 2.5.1.1.2 and Remark 2.5.2.1.2, we may view the characteristic relation
\[ \chi _{X}\webleft (-_{1},-_{2}\webright )\colon X\times X\to \{ \mathsf{t},\mathsf{f}\} \]
of $X$ as a decategorification of the $\textup{Hom}$ profunctor
\[ \textup{Hom}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )\colon \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\to \mathsf{Sets} \]
of a category $\mathcal{C}$.
