Definition lift_poset_counit : lift_poset_functor ⟹ constant_functor _ _ (monoidal_unit pointed_poset_sym_mon_closed_cat). Proof. use make_nat_trans. - simple refine (λ X, _ ,, _). + intro x. induction x as [ x | ]. * exact true. * exact false. + split. * abstract (intros x₁ x₂ p ; induction x₁ as [ x₁ | ], x₂ as [ x₂ | ] ; try (exact tt) ; cbn in * ; exact p). * abstract (cbn ; apply idpath). - abstract (intros X Y f ; use eq_mor_hset_struct ; intro x ; induction x as [ x | ] ; cbn ; apply idpath). Defined.
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.