We give a technique to construct a final coalgebra in which each element is a set of formulas of modal logic. The technique works for both the finite and the countable powerset functors. Starting with an injectively structured, corecursive algebra, we coinductively obtain a suitable subalgebra called the "co-founded part". We see---first with an example, and then in the general setting of modal logic on a dual adjunction---that modal theories form an injectively structured, corecursive algebra, so that this construction may be applied. We also obtain an initial algebra in a similar way.
We generalize the framework beyond Set to categories equipped with a suitable factorization system, and look at the examples of Poset and Set^op.