Final Coalgebras are Ideal Completions of Initial Algebras

For -continuous endofunctors of Set an ordering of a final coalgebra is exhibited which makes a CPO. Moreover, an initial algebra, considered as a canonical subobject of , has as its ideal completion. In more generality, for -continuous endofunctors of locally finitely presentable categories the analogous result holds: here the ordering is considered on the hom-sets for all finitely presentable objects .