Proposition is_lim_enriched_is_wedge (a : enriched_lim_cone) (w : C) : is_wedge (weighted_hom w) (is_lim_enriched_wedge_data a w). Proof. intros i j g ; cbn -[sym_mon_braiding]. rewrite !functor_id. rewrite postcomp_arr_id. use internal_funext. intros z h. rewrite !tensor_comp_r_id_r. rewrite !assoc'. unfold internal_pre_post_comp. rewrite !internal_beta. rewrite !tensor_id_id. rewrite id_left, id_right. rewrite !assoc. rewrite tensor_split. rewrite !assoc'. rewrite !(maponpaths (λ z, _ · z) (assoc _ _ _)). rewrite internal_beta. rewrite !assoc'. rewrite !(maponpaths (λ z, _ · z) (assoc _ _ _)). rewrite tensor_sym_mon_braiding. rewrite !assoc'. rewrite enriched_comp_postcomp_arr. rewrite !(maponpaths (λ z, _ · (_ · z)) (assoc _ _ _)). rewrite <- tensor_comp_id_r. rewrite enriched_lim_cone_commute. rewrite tensor_comp_id_r. rewrite !assoc'. rewrite !(maponpaths (λ z, _ · z) (assoc _ _ _)). rewrite <- tensor_sym_mon_braiding. rewrite !assoc. refine (!_). rewrite <- tensor_comp_mor. rewrite id_right. rewrite tensor_split. rewrite tensor_comp_id_l. rewrite !assoc'. apply maponpaths. rewrite internal_beta. rewrite tensor_split. rewrite !assoc'. do 2 apply maponpaths. rewrite !assoc. rewrite tensor_sym_mon_braiding. apply idpath. 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.