Lemma idtomor_to_transport {C : univalent_category} (TC TD : tensor C) (α : ∏ x y : C, (x ⊗_{TD} y) = (x ⊗_{TC} y)) (x1 x2 y1 y2 : C) (f : C⟦x1,x2⟧) (g : C⟦y1,y2⟧) : (idtomor _ _ (α x1 y1)) · (f ⊗^{TC} g) = (f ⊗^{TD} g) · (idtomor _ _ (α x2 y2)) -> transportf (λ x : C, C⟦x , x2 ⊗_{TC} y2⟧) (! α x1 y1) (f ⊗^{TC} g) = (transportf _ (α x2 y2) (f ⊗^{TD} g)). Proof. intro q. rewrite (! eq_idtoiso_idtomor _ _ _) in q. rewrite (! eq_idtoiso_idtomor _ _ _) in q. etrans. { apply (! idtoiso_precompose _ _ _ _ _ _). } etrans. { apply cancel_postcomposition. do 2 apply maponpaths. apply pathsinv0inv0. } etrans. { exact q. } apply idtoiso_postcompose. 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.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories. Import TensorNotation. Import TensorNotation.