The greatest fixed point of a set functor is proved to be (a) a metric completion and (b) a CPO-completion of finite iterations. For each (possibly infinitary) signature Σ the terminal Σ-coalgebra is thus proved to be the coalgebra of all Σ-labelled trees; this is the completion of the set of all such trees of finite depth. A set functor is presented which has a fixed point but does not have a greatest fixed point. A sufficient condition for the existence of a greatest fixed point is proved: the existence of two fixed points of successor cardinalities. © 1995.
Adámek, J., & Koubek, V. (1995). On the greatest fixed point of a set functor. Theoretical Computer Science, 150(1), 57–75. https://doi.org/10.1016/0304-3975(95)00011-K