proposition: Proposition eq_is_democratic {C : full_comp_cat} {D₁ D₂ : is_democratic C} (p : ∏ (Γ : C), is_democratic_ty D₁ Γ = is_democratic_ty D₂ Γ) (q : ∏ (Γ : C), is_democratic_iso D₁ Γ · comp_cat_comp_mor (idtoiso_disp (idpath _) (p Γ)) = is_democratic_iso D₂ Γ) : D₁ = D₂.
proof: Proof. use funextsec ; intro Γ. use total2_paths_f. - exact (p Γ). - use subtypePath. { intro. apply isaprop_is_z_isomorphism. } unfold z_iso. rewrite pr1_transportf. rewrite transportf_dem. exact (q Γ). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.FullyFaithfulDispFunctor UniMath.CategoryTheory.Limits.Terminal UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.DisplayedBicats.Examples.FullSub UniMath.Bicategories.ComprehensionCat.BicatOfCompCat UniMath.Bicategories.ComprehensionCat.CompCatNotations