Let $X$ be a set.
We have a natural identification1
for an extension of this correspondence to “relative monads in $\boldsymbol {\mathsf{Rel}}$”. A monad in $\boldsymbol {\mathsf{Rel}}$ on $X$ consists of a relation $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X$ together with maps
making the diagrams
For each $x,z\in X$, if there exists some $y\in Y$ such that $x\sim _{R}y$ and $y\sim _{R}z$, then $x\sim _{R}z$.
For each $x\in X$, we have $x\sim _{R}x$.
These are exactly the requirements for $R$ to be a preorder (). Conversely, any preorder $\preceq $ gives rise to a pair of maps $\mu _{\preceq }$ and $\eta _{\preceq }$, forming a monad on $X$.
Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.
The codensity monad $\operatorname {\mathrm{Ran}}_{R}(R)\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ is given by
on $B$ obtained by declaring $b\preceq _{\operatorname {\mathrm{Ran}}_{R}(R)}b'$ iff the following equivalent conditions are satisfied:
The dual codensity monad $\operatorname {\mathrm{Rift}}_{R}(R)\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ is given by
on $A$ obtained by declaring $a\preceq _{\operatorname {\mathrm{Rift}}_{R}(R)}a'$ iff the following equivalent conditions are satisfied:
