Proposition is_mon_nat_trans_lift_poset_dupl : is_mon_nat_trans lax_monoidal_lift_poset_comonad (comp_fmonoidal_lax lax_monoidal_lift_poset_comonad lax_monoidal_lift_poset_comonad) lift_poset_dupl. Proof. split. - intros X Y. use eq_mor_hset_struct. use setquotunivprop' ; [ intro ; apply setproperty | ]. intros xy. induction xy as [ x y ]. induction x as [ x | ] ; induction y as [ y | ]. + apply idpath. + apply idpath. + apply idpath. + apply idpath. - use eq_mor_hset_struct. intro x. induction x ; cbn ; apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructuresSmashProduct. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.PointedPosetStrict. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.FunctorCategories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Require Import UniMath.CategoryTheory.Monoidal.Examples.SmashProductMonoidal. Require Import UniMath.CategoryTheory.Monoidal.Examples.PosetsMonoidal. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.CategoryOfPosets. Require Import UniMath.CategoryTheory.Monads.Comonads. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Category.