proposition: Lemma uniqueness_algebraic_theory_data_disp_cat (d' : D (lim L)) (cone_out : ∏ (j : vertex J), d' -->[limOut L j] (pr2 (dob d j))) : d' = tip_algebraic_theory_data_disp_cat.
proof: Proof. apply pathsdirprod. - do 2 (apply funextsec; intro). apply subtypePairEquality. { intro. repeat (apply impred_isaprop; intro). apply setproperty. } apply funextsec. intro. exact (pr1 (cone_out _) _ _). - do 4 (apply funextsec; intro). apply subtypePairEquality. { intro. repeat (apply impred_isaprop; intro). apply setproperty. } apply funextsec. intro. exact (pr2 (cone_out _) _ _ _ _). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Limits UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.AlgebraicTheories.IndexedSetCategory UniMath.AlgebraicTheories.AlgebraicTheoryCategoryCore UniMath.AlgebraicTheories.AlgebraicTheories UniMath.AlgebraicTheories.AlgebraicTheoryMorphisms