Proposition enriched_weighted_to_conical_is_conical_lim_unique : isaprop (∑ (g : v --> E ⦃ w, a ⦄), ∏ (i : I), g · postcomp_arr E w (enriched_to_arr E (enriched_lim_cone_pr E D W a i)) = fs i). Proof. use invproofirrelevance. intros φ₁ φ₂. use subtypePath. { intro. use impred ; intro. apply homset_property. } use (mor_to_end_eq _ (pr2 a w)). intros i ; cbn. use internal_funext. intros z h. rewrite !tensor_comp_r_id_r. rewrite !assoc'. rewrite !internal_beta. rewrite !assoc. rewrite <- !tensor_comp_mor. rewrite !id_right. rewrite !tensor_sym_mon_braiding. rewrite !assoc'. apply maponpaths. rewrite !tensor_comp_r_id_l. rewrite !assoc'. apply maponpaths. pose (p := maponpaths (λ z, mon_lunitor _ · z) (pr2 φ₁ i @ !(pr2 φ₂ i))). cbn in p. unfold postcomp_arr in p. rewrite !enriched_from_to_arr in p. rewrite !assoc' in p. rewrite !(maponpaths (λ z, _ · z) (assoc _ _ _)) in p. rewrite !tensor_linvunitor in p. rewrite !assoc in p. rewrite !mon_lunitor_linvunitor in p. rewrite !id_left in p. rewrite <- !tensor_split in p. exact p. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Limits.Ends. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.SelfEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedConicalLimits. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedPowers. Import MonoidalNotations.