proposition: Proposition is_mon_nat_trans_lift_poset_extract : is_mon_nat_trans lax_monoidal_lift_poset_comonad (identity_fmonoidal _) lift_poset_extract.
proof: 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 | ]. + use iscompsetquotpr ; cbn. apply hinhpr. apply inl. apply idpath. + use iscompsetquotpr ; cbn. apply hinhpr. apply inr. split. * apply inl. apply idpath. * apply inr. apply idpath. + use iscompsetquotpr ; cbn. apply hinhpr. apply inr. split. * apply inl. apply idpath. * apply inl. apply idpath. + use iscompsetquotpr ; cbn. apply hinhpr. apply inr. split. * apply inl. apply idpath. * apply inr. apply idpath. - use eq_mor_hset_struct. intro x. induction x ; cbn ; 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