We summarise the analogies between functions relations and profunctors in the following table:
| Set Theory | Category Theory |
|---|---|
| Relation $R\colon X\times Y\to \{ \mathsf{t},\mathsf{f}\} $ | Profunctor $\mathfrak {p}\colon \mathcal{D}^{\mathsf{op}}\times \mathcal{C}\to \mathsf{Sets}$ |
| Relation $R\colon X\to \mathcal{P}\webleft (Y\webright )$ | Profunctor $\mathfrak {p}\colon \mathcal{C}\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$ |
| Relation as a cocontinuous morphism of posets $R\colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (Y\webright ),\subset \webright )$ | Profunctor as a colimit-preserving functor $\mathfrak {p}\colon \mathsf{PSh}\webleft (\mathcal{C}\webright )\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$ |
