Lemma TransportedAssocRight : nat_z_iso (HHH ∙ assoc_right TD) (assoc_right TC ∙ H). Proof. use nat_z_iso_comp. 2: apply nat_z_iso_functor_comp_assoc. use nat_z_iso_comp. 2: { use post_whisker_nat_z_iso. 2: exact unassoc_commutes. } use nat_z_iso_comp. 2: apply (nat_z_iso_inv (nat_z_iso_functor_comp_assoc _ _ _)). use nat_z_iso_comp. 3: apply nat_z_iso_functor_comp_assoc. apply pre_whisker_nat_z_iso. use nat_z_iso_comp. 3: apply nat_z_iso_functor_comp_assoc. use nat_z_iso_comp. 3: { apply pre_whisker_nat_z_iso. apply TransportedTensorComm. } use nat_z_iso_comp. 3: apply (nat_z_iso_inv (nat_z_iso_functor_comp_assoc _ _ _)). use nat_z_iso_comp. 2: apply nat_z_iso_functor_comp_assoc. apply post_whisker_nat_z_iso. use product_of_commuting_squares. { apply (make_nat_z_iso _ _ _ (is_nat_z_iso_nat_trans_id H)). } apply TransportedTensorComm. Defined.
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.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.catiso. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.PrecompEquivalence. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.TotalCategoryFacts. Require Import UniMath.CategoryTheory.Monoidal.RezkCompletion.LiftedTensor. Require Import UniMath.CategoryTheory.Monoidal.RezkCompletion.LiftedTensorUnit.