An axiomatics for categories of coalgebras

We give an axiomatic account of what structure on a category C and an endofunctor H on C yield similar structure on the category H ?Coalg of H-coalgebras. We give conditions under which completeness, cocompleteness, symmetric monoidal closed structure, local presentability, and subobject classifiers lift. Our proof of the latter uses a general result about the existance of a subobject classifier in a category containing a small dense subcategory. Our leading example has C = Set with H the endofunctor for which a coalgebra is a finitely branching (labelled) transition system. We explain that example in detail.