proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.EnrichedCats.Enrichment UniMath.CategoryTheory.EnrichedCats.Examples.SelfEnriched UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedInitial UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedBinaryCoproducts UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCoproducts UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCoequalizers UniMath.CategoryTheory.EnrichedCats.Colimits.EnrichedCopowers UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.Closed UniMath.CategoryTheory.Limits.Preservation UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Equalizers