proposition: Lemma tensor_eqi''_equiv_tensor_eqi''' {C : univalent_category} (TC TD : tensor C) : tensor_eqi''' TC TD ≃ tensor_eqi'' TC TD.
proof: Proof. apply weqfibtototal. intro. repeat (apply weqonsecfibers ; intro). use weq_iso. - intro q. apply pathsinv0. etrans. { apply maponpaths. apply maponpaths. exact (! q). } rewrite (! independent_transportation_commutes _ _ _). rewrite transport_f_f. rewrite pathsinv0r. rewrite idpath_transportf. rewrite transport_f_f. rewrite pathsinv0r. apply idpath_transportf. - intro q. etrans. { apply maponpaths. apply maponpaths. exact q. } rewrite (! independent_transportation_commutes _ _ _). rewrite transport_f_f. rewrite pathsinv0l. rewrite idpath_transportf. rewrite transport_f_f. rewrite pathsinv0l. apply idpath_transportf. - intro q. apply homset_property. - intro q. 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