Let $R$ be a relation on $A$.

  1. Adjointness. We have an adjunction
    witnessed by a bijection of sets
    \[ \mathbf{Rel}^{\mathsf{refl}}\webleft (R^{\mathrm{refl}},S\webright ) \cong \mathbf{Rel}\webleft (R,S\webright ), \]

    natural in $R\in \text{Obj}\webleft (\mathbf{Rel}^{\mathsf{refl}}\webleft (A,A\webright )\webright )$ and $S\in \text{Obj}\webleft (\mathbf{Rel}\webleft (A,A\webright )\webright )$.

  2. The Reflexive Closure of a Reflexive Relation. If $R$ is reflexive, then $R^{\mathrm{refl}}=R$.
  3. Idempotency. We have
    \[ \webleft (R^{\mathrm{refl}}\webright )^{\mathrm{refl}} = R^{\mathrm{refl}}. \]
  4. Interaction With Inverses. We have
  5. Interaction With Composition. We have


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