Let $R$ be a relation on $A$.
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Adjointness. We have an adjunction witnessed by a bijection of sets
\[ \mathbf{Rel}^{\text{eq}}\webleft (R^{\mathrm{eq}},S\webright ) \cong \mathbf{Rel}\webleft (R,S\webright ), \]natural in $R\in \text{Obj}\webleft (\mathbf{Rel}^{\text{eq}}\webleft (A,B\webright )\webright )$ and $S\in \text{Obj}\webleft (\mathbf{Rel}\webleft (A,B\webright )\webright )$.
- The Equivalence Closure of an Equivalence Relation. If $R$ is an equivalence relation, then $R^{\mathrm{eq}}=R$.
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Idempotency. We have
\[ \webleft (R^{\mathrm{eq}}\webright )^{\mathrm{eq}} = R^{\mathrm{eq}}. \]
