Proposition power_from_lim_wedge_inv_1 : is_power_enriched_map E v x power_from_lim_cone w · mor_to_end (weighted_hom E D W w) (enriched_lim_cone_is_lim E _ _ a w) power_from_lim_wedge = identity _. Proof. use (mor_to_end_eq _ (pr2 a w)). intro i. rewrite !assoc'. etrans. { apply maponpaths. exact (mor_to_end_comm _ (pr2 a w) power_from_lim_wedge i). } refine (id_right _ @ _ @ !(id_left _)). use internal_funext. intros z h. rewrite tensor_split. rewrite !assoc'. unfold is_power_enriched_map. rewrite internal_beta. refine (!_). rewrite tensor_split. rewrite !assoc'. apply maponpaths. etrans. { apply internal_beta. } rewrite !assoc. induction i. 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.