Inverse Relation Functor is Contravariant Functor

Theorem
Let $\mathbf{Rel}$ be the category of relations.

Let $C: \mathbf{Rel} \to \mathbf{Rel}$ be the inverse relation functor.

Then $C$ is a contravariant functor.

Proof
For any set $X$. we have:

For any two relations $\mathcal R \subseteq X \times Y$ and $\mathcal S \subseteq Y \times Z$, we have:

Hence $C$ is shown to be a contravariant functor.