Proposition nat_trans_enrichment_via_comp {V : monoidal_cat} {C₁ C₂ : category} {F G : C₁ ⟶ C₂} (τ : nat_trans_data F G) {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} {EF : functor_enrichment F E₁ E₂} {EG : functor_enrichment G E₁ E₂} (p : ∏ (x y : C₁), EG x y · precomp_arr E₂ (G y) (τ x) = EF x y · postcomp_arr E₂ (F x) (τ y)) : nat_trans_enrichment τ EF EG. Proof. intros x y. refine (_ @ p x y @ _) ; unfold postcomp_arr, precomp_arr. - rewrite !assoc. apply maponpaths_2. rewrite tensor_rinvunitor. rewrite !assoc'. rewrite <- tensor_split'. apply idpath. - rewrite !assoc. apply maponpaths_2. rewrite tensor_linvunitor. rewrite !assoc'. rewrite <- tensor_split. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.whiskering.