We have a natural identification
\[ \left\{ \begin{gathered} \text{Comonads in}\\ \text{$\boldsymbol {\mathsf{Rel}}$ on $A$} \end{gathered} \right\} \cong \left\{ \text{Subsets of $A$}\right\} . \]
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1.
For each $a,b\in A$, if $a\sim _{R}b$, then there exists some $k\in A$ such that $a\sim _{R}k$ and $k\sim _{R}b$.
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2.
For each $a,b\in A$, if $a\sim _{R}b$, then $a=b$.
8.4.5 Internal Comonads
Let $A$ be a set.
Proof of Proposition 8.4.5.1.1.
A comonad in $\boldsymbol {\mathsf{Rel}}$ on $A$ consists of a relation $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ together with maps
\begin{align*} \Delta _{R} & \colon R \subset R\mathbin {\diamond }R,\\ \epsilon _{R} & \colon R \subset \chi _{A} \end{align*}
making the diagrams
Taking $k=b$ in the first condition above shows it to be trivially satisfied, while the second condition implies $R\subset \Delta _{A}$, i.e. $R$ must be a subset of $A$. Conversely, any subset $U$ of $A$ satisfies $U\subset \Delta _{A}$, defining a comonad as above.
