Proposition tensor_enriched_precat_id_ax : enriched_id_ax tensor_enriched_precat_data. Proof. intros x y. induction x as [ x₁ x₂ ]. induction y as [ y₁ y₂ ]. split ; cbn. - rewrite !assoc. rewrite tensor_comp_r_id_r. rewrite <- tensor_id_id. etrans. { apply maponpaths_2. rewrite !assoc'. apply maponpaths. rewrite <- naturality_inner_swap. apply idpath. } rewrite !assoc'. etrans. { do 2 apply maponpaths. rewrite <- tensor_comp_mor. rewrite <- !enrichment_id_left. apply idpath. } rewrite <- precompose_inner_swap_with_lunitors_on_right. etrans. { apply maponpaths. rewrite !assoc. do 2 apply maponpaths_2. apply inner_swap_inv. } rewrite id_left. rewrite tensor_mor_right. rewrite !assoc. rewrite tensor_id_id. etrans. { apply maponpaths_2. refine (!_). apply tensor_comp_id_r. } rewrite mon_linvunitor_lunitor. rewrite tensor_id_id. apply id_left. - rewrite !assoc. rewrite tensor_comp_l_id_r. rewrite <- tensor_id_id. etrans. { apply maponpaths_2. rewrite !assoc'. apply maponpaths. rewrite <- naturality_inner_swap. apply idpath. } rewrite !assoc'. etrans. { do 2 apply maponpaths. rewrite <- tensor_comp_mor. rewrite <- !enrichment_id_right. apply idpath. } rewrite <- precompose_inner_swap_with_lunitors_and_runitor. etrans. { apply maponpaths. rewrite !assoc. do 2 apply maponpaths_2. apply inner_swap_inv. } rewrite id_left. rewrite tensor_mor_left. rewrite !assoc. rewrite tensor_id_id. etrans. { apply maponpaths_2. refine (!_). apply tensor_comp_id_l. } rewrite mon_linvunitor_lunitor. rewrite tensor_id_id. apply id_left. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.Enriched.Enriched. Require Import UniMath.CategoryTheory.EnrichedCats.Enriched.EnrichmentEquiv. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal. Import MonoidalNotations.