 Category of relations

For a regular category $\mathcal{C}$, the category $\mathbf{Rel}(\mathcal{C})$ has the objects of $\mathcal{C}$ and subobjects of $A\times B$ as relations $A⇝B$. Composition takes the image of the pullback expressing existential quantification over the middle object.

For $\mathcal{C}=\mathbf{Set}$ this is the category of relations (sets); the general construction also applies to other regular categories.