Category of Relations is Category

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

Then $\mathbf{Rel}$ is a metacategory.


Let us verify the axioms $(C1)$ up to $(C3)$ for a metacategory.

We have already defined the composition of morphisms.

For any set $X$, we have the diagonal relation $\operatorname{id}_X$.

By Diagonal Relation is Left Identity and Diagonal Relation is Right Identity it follows that this is the identity morphism for $X$.

Finally by Composition of Relations is Associative, the associative property is satisfied.

Hence $\mathbf{Rel}$ is a metacategory.