Lemma TransportedAssocRightOnOb (x y z : C) : TransportedAssocRight ((x,y),z) = # TD (# H (id x) #, TransportedTensorComm Duniv H_eso H_ff TC (y, z)) · TransportedTensorComm Duniv H_eso H_ff TC (x, TC (y,z)). Proof. cbn. rewrite ! id_left. rewrite ! id_right. rewrite ! functor_id. rewrite ! assoc. apply maponpaths_2. fold (TransportedTensor Duniv H_eso H_ff). etrans. { apply (! functor_comp TD _ _). } apply maponpaths. rewrite <- binprod_comp. rewrite id_right. apply maponpaths. etrans. { apply maponpaths_2. apply (functor_id TD). } apply id_left. Qed.
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.