proposition: Lemma tensor_iso_equiv_tensor_eq {C : univalent_category} (TC TD : tensor C) : tensor_eq TC TD ≃ tensor_iso TC TD.
proof: Proof. use weqtotal2. - apply weqonsecfibers ; intro x ; apply weqonsecfibers ; intro y. use weq_iso. + apply idtoiso. + apply isotoid. apply univalent_category_is_univalent. + intro. apply isotoid_idtoiso. + intro. apply idtoiso_isotoid. - intro iso_tob. repeat (apply weqonsecfibers ; intro). use weq_iso. + intro p. cbn in p. rewrite (! eq_idtoiso_idtomor _ _ _) in p. rewrite (! eq_idtoiso_idtomor _ _ _) in p. apply p. + intro p. etrans. { apply cancel_postcomposition. apply (! eq_idtoiso_idtomor _ _ _). } etrans. { apply p. } apply cancel_precomposition. apply eq_idtoiso_idtomor. + intro ; apply homset_property. + intro ; apply homset_property. Defined.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.DisplayedBicats.DispInvertibles UniMath.Bicategories.Core.Univalence UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories