Let $F:Set\to Set$ be a functor. An $F$-coalgebra is a pair $\mathcal{A}=(A,\alpha)$ where $\alpha:A\to F(A)$ is arbitrary map. Given $F$-coalgebras $\mathcal{A}=(A,\alpha)$ and $\mathcal{B}=(B,\beta)$, a homomorphism $\varphi:A\to B$ is a map $\varphi:A\to B$ which makes the following diagram commute:

It is well known that if **terminal** (final) **coalgebra** exists then a structure map of the coalgebra should be isomorphism. It is easy to see that the category $Set_{\mathcal{P}}$ of all covariant powerset-coalgebras has no terminal object (since obviously there is no such set $A$ that $A\cong \mathcal{P}(A)$ holds).

The problem of **existence of limits** for coalgebras corresponding to general and particular functors was considered by various authors, the last I was able to find, is "*H. Peter Gumm & Tobias Schroder, Products of coalgebras, Algebra Universalis 46 (2001) 163 – 185.*" where the autors generalized known for the moment results.

In particular Gumm & Schroder prove the **category of $F$-coalgebras is complete**, that is products and equalizers exist, **provided that the functor $F$ is bounded or preserves mono sources**. This generalizes and simplifies a result of Worrell, for a functor $F:Set\to Set$ J. Worrell was able to show that $Set_F$ is complete (where $Set_{F}$ is the category of all $F$-coalgebras), that is products and equalizers exist, provided the functor $F$ weakly preserves pullbacks and $F$ is bounded.

Since $P_{\omega}$, the finite powerset functor is bounded (by $\omega$), hence the class of all coalgebras of $P_{\omega}$, is complete. But I am not able to find any other further development toward powerset coalgebras.

**Thus I am looking for ( reference on) further development of the theory,
in particular are there known:**

Q.1.What are other known possible complete subclasses (subcategories) of powerset coalgebras?

Q.2.What are examples of incomplete subclasses of powerset coalgebras where construction (in a uniform way??) of existing products is known?