Proposition enriched_rezk_completion_ump_comm_enrichment : nat_trans_enrichment enriched_rezk_completion_ump_nat_z_iso EG (functor_comp_enrichment EF enriched_rezk_completion_ump_functor_enrichment). Proof. use nat_trans_enrichment_via_comp. intros x y ; cbn. rewrite (enriched_rezk_completion_ump_functor_enrichment_eq (identity_z_iso (F x)) (identity_z_iso (F y))) ; cbn. rewrite precomp_arr_id, postcomp_arr_id. rewrite !id_left. rewrite !assoc. etrans. { do 4 apply maponpaths_2. exact (z_iso_inv_after_z_iso (fully_faithful_enriched_functor_z_iso HF₂ x y)). } rewrite id_left. rewrite !assoc'. apply maponpaths. rewrite <- precomp_postcomp_arr. rewrite !assoc. rewrite <- precomp_arr_comp. rewrite z_iso_inv_after_z_iso. rewrite precomp_arr_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.whiskering. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.EnrichedCats.FullyFaithful. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.FunctorCategory. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.Precomposition. Require Import UniMath.CategoryTheory.Monoidal.Categories.