The internal Hom of $\mathsf{Rel}$ is the functor
\[ \mathrm{Rel}\colon \mathsf{Rel}^{\mathsf{op}}\times \mathsf{Rel}\to \mathsf{Rel} \]
defined
- On objects by sending $A,B\in \text{Obj}\webleft (\mathsf{Rel}\webright )$ to the set $\mathrm{Rel}\webleft (A,B\webright )$ of
of . - On morphisms by pre/post-composition defined as in Chapter 7: Constructions With Relations, Definition 7.3.12.1.1.
