Coalgebraic description of converse of a binary relation on a set

Question

A binary relation on a set is a coalgebra for the powerset endofunctor on the category of sets. In this coalgebraic setting, how do you construct or characterize the coalgebra which represents the converse relation of a given relation?

Answer 1

Suppose $\mathcal{P} : \mathtt{Set} \to \mathtt{Set}$ is the power set endofunctor. Then, $\mathcal{P}$-coalgebras are in bijective correspondence with binary relations.

Let $r : X \to \mathcal{P}(X)$ is a $\mathcal{P}$-coalgebra. Define the binary relation $$R = \big\{ (x,y) \;\big|\; y \in r(x) \subset X \big\} \subset X \times X.$$ Conversely, given any binary relation $S \subset X \times X$, construct the $\mathcal{P}$-coalgebra $s : X \to \mathcal{P}(X)$ by $$s(x) = \{ y \in X \,|\, (x,y) \in S\}.$$

Clearly, these are inverses to each other.

Answer 2

(to answer my own question) I was looking for something like this:

Let $\gamma: X \to PY$ represent the relation $R \subseteq X \times Y$ where $P$ is the covariant powerset functor. Let $P^{\!\ast}$ be the contravariant powerset functor. Then the coalgebra representing the converse $\breve{R} \subseteq Y \times X$ is the following composite. $$Y \stackrel{\{.\}}{\to} PY \stackrel{\subseteq}{\to} P^{\!\ast} PY \stackrel{P^{\!\ast} \gamma}{\to} P^{\!\ast} X = PX$$