Lemma LiftPreservesPostTensor : nat_z_iso (H ∙ I_posttensor TD (H I)) (I_posttensor TC I ∙ H). Proof. use nat_z_iso_comp. 2: { apply pre_whisker_nat_z_iso. apply tensor_after_pair_with_object_right. } use nat_z_iso_comp. 3: { apply nat_z_iso_inv. use post_whisker_nat_z_iso. 2: apply tensor_after_pair_with_object_right. } 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. exact (TransportedTensorComm Duniv H_eso H_ff TC). } use nat_z_iso_comp. 2: { apply nat_z_iso_functor_comp_assoc. } use nat_z_iso_comp. 3: { apply (nat_z_iso_inv (nat_z_iso_functor_comp_assoc _ _ _)). } apply post_whisker_nat_z_iso. apply PairingWithObjectCommutesRight. 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.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.PrecompEquivalence. Require Import UniMath.CategoryTheory.catiso. 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.