proposition: 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: 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.
imports: UniMath.MoreFoundations.All UniMath.OrderTheory.Posets UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits UniMath.CategoryTheory.DisplayedCats.Structures.StructuresSmashProduct UniMath.CategoryTheory.DisplayedCats.Examples.PointedPosetStrict UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.FunctorCategories UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.Closed UniMath.CategoryTheory.Monoidal.Examples.SmashProductMonoidal UniMath.CategoryTheory.Monoidal.Examples.PosetsMonoidal UniMath.CategoryTheory.DisplayedCats.Examples.CategoryOfPosets UniMath.CategoryTheory.Monads.Comonads UniMath.CategoryTheory.Monoidal.Comonoids.Category