Product / Co-product in category of sets and relations (Rel)

Question

What is the co-product and product in category of sets and relations.

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Answer 1

Both the product and coproduct of a family of sets $X_i$ are given as the disjoint union $\coprod X_i$ of sets together with the relations $$\left\{(a, a) \in \left(\coprod X_i\right) \times X_j \ \middle| \ \forall a \in X_j\right\}$$ and $$\left\{(a, a) \in X_j \times \left(\coprod X_i\right) \ \middle| \ \forall a \in X_j\right\}.$$