Definition all_invertible_map1cells {F G : ps_base C D} {η ε : F --> G} (m : invertible_2cell η ε) {F₁ : map1cells_disp_bicat F} {G₁ : map1cells_disp_bicat G} {η₁ : F₁ -->[ η ] G₁} {ε₁ : F₁ -->[ ε ] G₁} (m₁ : η₁ ==>[ m ] ε₁) : is_disp_invertible_2cell m m₁. Proof. use tpair. - exact (all_invertible_map1cells_inv m m₁). - split ; repeat (apply funextsec ; intro) ; apply D. Qed. Section AllInvertible2CellToDispAdjEquiv. Variable (F₀ : ps_base C D) (F₁ F₁' : map1cells_disp_bicat F₀) (η : (∏ (X Y : C) (f : X --> Y), invertible_2cell (F₁ X Y f) (F₁' X Y f))). Local Definition all_invertible_left_adj : F₁ -->[ internal_adjoint_equivalence_identity F₀] F₁'. Proof. intros X Y f ; cbn. use tpair. - exact (lunitor _ • (η X Y f)^-1 • rinvunitor _). - cbn. is_iso. Defined.
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.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Display.Base. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.