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