proposition: Lemma sigma_lim_arrow_unique (t : ∑ x, is_cone_mor d'_cone (make_cone _ forms_cone_sigma_disp_cat) x) : t = sigma_lim_arrow ,, sigma_lim_arrow_commutes.
proof: Proof. use subtypePairEquality. { intro. apply impred_isaprop. intro. apply homset_property. } pose (f' := (# sigma_to_E_total_functor)%cat (pr1 t)). pose (Hf' := λ u, maponpaths (# sigma_to_E_total_functor)%cat (pr2 t u)). pose (uniq := limArrowUnique d_cone _ e_cone f' Hf'). exact (maponpaths (# E_to_sigma_total_functor)%cat uniq). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Limits