On universal categories of coalgebras

A category ${\mathcal{K}}$ is called universal if for every accessible functor F : Set → Set the category of all F-coalgebras and the category of all F-algebras can be fully embedded into ${\mathcal{K}}$ . We prove that for a functor G preserving intersections, the category Coalg G of all G-coalgebras is universal unless the functor G is linear, that is, of the form GX = X × A + B for some fixed sets A and B. Other types of universality are also investigated.