Let $R$ be a relation on $A$.
-
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 $S\in \text{Obj}\webleft (\mathbf{Rel}\webleft (A,A\webright )\webright )$.
- The Reflexive Closure of a Reflexive Relation. If $R$ is reflexive, then $R^{\mathrm{refl}}=R$.
-
Idempotency. We have
\[ \webleft (R^{\mathrm{refl}}\webright )^{\mathrm{refl}} = R^{\mathrm{refl}}. \]
-
Interaction With Inverses. We have
-
Interaction With Composition. We have
