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

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


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