proposition: Lemma tensor_iso_disp_eq_equiv {C : bicat_of_univ_cats} (TC TD : bidisp_tensor_disp_bicat C) : tensor_iso TC TD ≃ disp_adjoint_equivalence (idtoiso_2_0 C C (idpath C)) TC TD.
proof: Proof. use weq_iso. - intro ti. apply (tensoriso_to_disp_eq TC TD ti). - intro dae. exact (disp_eq_to_tensoriso TC TD dae). - intro i. induction i. use subtypePath. + intro f. repeat (apply impred_isaprop ; intro). apply univalent_category_has_homsets. + repeat (apply funextsec ; intro). use subtypePath. * intro f. apply isaprop_is_z_isomorphism. * repeat (apply funextsec ; intro). apply idpath. - intro dae. use subtypePath. + intro f. apply isaprop_disp_left_adjoint_equivalence. * apply univalent_cat_is_univalent_2_1. * apply bidisp_tensor_disp_bicat_is_locally_univalent. + apply idpath. 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