proposition: Lemma dispadjequiv_to_iso (C : bicat_of_univ_cats) (IC ID : bidisp_unit_disp_bicat C) : disp_adjoint_equivalence (idtoiso_2_0 C C (idpath C)) IC ID -> z_iso IC ID.
proof: Proof. intro equalunits. induction equalunits as [f adj]. induction adj as [data [ladj_ax weq_ax]]. induction data as [finv [dunit dcounit]]. cbn in *. unfold functor_identity in dunit. unfold nat_trans_id in dunit. cbn in dunit. exists finv. exists f. split. - etrans. { apply (pathsinv0 dunit). } apply id_left. - apply pathsinv0. etrans. { apply (pathsinv0 dcounit). } apply id_right. 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.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories