Let $R$ be a preorder on $X$, viewed as an internal monad on $X$ via Proposition 8.5.4.1.1.
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Eilenberg–Moore Objects in $\boldsymbol {\mathsf{Rel}}$. The Eilenberg–Moore object for $R$ exists iff it is an equivalence relation, in which case it is the quotient $X/\mathord {\sim }_{R}$ of $X$ by $R$.
8.5.8 Eilenberg–Moore and Kleisli Objects
Let $X$ be a set.
