Proposition is_strict_and_monotone_lift_poset_comult_map (X : pointed_poset_sym_mon_closed_cat) : is_strict_and_monotone (pr2 (lift_poset_comonad X)) (pr2 (lift_poset_comonad X ∧* lift_poset_comonad X)) (@lift_poset_comult_map X). Proof. split. - intros x₁ x₂ p. induction x₁ as [ x₁ | ], x₂ as [ x₂ | ] ; cbn in *. + refine (hinhpr (inr _)). exact (p ,, p). + exact (fromempty p). + refine (hinhpr (inr _)). exact (tt ,, tt). + refine (hinhpr (inr _)). exact (tt ,, tt). - 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.