Proposition self_enrichment_is_copower_eq_2 (w : V) : internal_uncurry v₁ v₂ w · is_copower_enriched_map (self_enrichment V) v₁ v₂ self_enrichment_copower_cocone w = identity _. Proof. use internal_funext ; cbn. intros a₁ h₁. rewrite tensor_comp_r_id_r. rewrite !assoc'. unfold is_copower_enriched_map ; cbn. rewrite internal_beta. use internal_funext ; cbn. intros a₂ h₂. rewrite !tensor_comp_r_id_r. rewrite !assoc'. unfold internal_comp. rewrite internal_beta. etrans. { apply maponpaths. 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. rewrite tensor_lassociator. rewrite !assoc'. etrans. { apply maponpaths. rewrite tensor_split. rewrite !assoc'. unfold internal_uncurry. rewrite internal_beta. rewrite !assoc. rewrite tensor_rassociator. rewrite !assoc'. apply idpath. } rewrite !assoc. rewrite mon_lassociator_rassociator. rewrite id_left. apply idpath. 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.