Definition final_coalg_to_mhss_alt : mhss Mon_V H θ. Proof. exists t. exists η. exists τ. intros Z f. simple refine (iscontrretract _ _ _ (cia_from_final_coalgebra CP I_H _ isTerminalνH _ (final_coalg_to_mhss_equation_morphism f))). - intros [h Hyp]. exists h. apply final_coalg_to_mhss_equation_morphism_has_equivalent_characteristic_formula. exact Hyp. - intros [h Hyp]. exists h. apply final_coalg_to_mhss_equation_morphism_has_equivalent_characteristic_formula. exact Hyp. - intros [h Hyp]. use total2_paths_f. + apply idpath. + apply isaprop_mbracket_property_parts. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.FunctorCoalgebras. Require Import UniMath.CategoryTheory.coslicecat. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.Chains. Require Import UniMath.CategoryTheory.Chains.Adamek. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.CompletelyIterativeAlgebras. Require Import UniMath.CategoryTheory.Chains.OmegaCocontFunctors. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids. Require Import UniMath.CategoryTheory.Actegories.Actegories. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories. Require Import UniMath.CategoryTheory.Actegories.MorphismsOfActegories. Require Import UniMath.CategoryTheory.Actegories.CoproductsInActegories. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects. Require Import UniMath.SubstitutionSystems.GeneralizedSubstitutionSystems. Require Import UniMath.SubstitutionSystems.ActionScenarioForGenMendlerIteration_alt. Require Import UniMath.SubstitutionSystems.SigmaMonoids. Import BifunctorNotations. Import MonoidalNotations. Theorem SigmaMonoidFromInitialAlgebra_is_initial : isInitial _ σ. Definition SigmaMonoidFromInitialAlgebraInitial : Initial (SigmaMonoid θ) := σ,,SigmaMonoidFromInitialAlgebra_is_initial.