Definition fmap_directed {P Q} (f : posetmorphism P Q) : Directed_family P -> Directed_family Q. Proof. apply funfibtototal; intros I. use bandfmap. - apply (fun C => f ∘ C). - apply fmap_is_directed. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.OrderedSets.WellOrderedSets. Require Import UniMath.OrderTheory.DCPOs.AlternativeDefinitions.Dcpo. Require Import UniMath.OrderTheory.Posets.Subposets.