Category of relations ($\mathbf{Rel}$)

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.

The graph of $f:A\to B$ is the relation represented by $(1_A,f):A\hookrightarrow A\times B$. Its converse is right adjoint to it in the pointwise order on relations.