Proposition self_enrichment_is_copower_eq_1 (w : V) : is_copower_enriched_map (self_enrichment V) v₁ v₂ self_enrichment_copower_cocone w · internal_uncurry v₁ v₂ w = identity _. Proof. cbn. use internal_funext. intros a h. rewrite tensor_comp_r_id_r. rewrite !assoc'. unfold internal_uncurry. rewrite internal_beta. unfold is_copower_enriched_map. rewrite tensor_split. rewrite <- tensor_id_id. etrans. { rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_rassociator. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_r. rewrite internal_beta ; cbn. rewrite tensor_comp_id_r. rewrite !assoc'. unfold internal_comp. rewrite internal_beta. rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_l. unfold internal_pair. rewrite internal_beta. rewrite tensor_id_id. apply id_left. } rewrite !assoc. apply maponpaths_2. rewrite !assoc'. rewrite mon_rassociator_lassociator. apply id_right. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.SelfEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedInitial. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedBinaryCoproducts. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCoproducts. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCoequalizers. Require Import UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCopowers. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Import MonoidalNotations.