On quantale-enriched categories the Hausdorff functor has no terminal coalgebra, but on enriched compact Hausdorff spaces it preserves codirected limits, making categories of Hausdorff polynomial coalgebras complete.
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Hausdorff coalgebras
On quantale-enriched categories the Hausdorff functor has no terminal coalgebra, but on enriched compact Hausdorff spaces it preserves codirected limits, making categories of Hausdorff polynomial coalgebras complete.