Definition discrete_fiber_runitor {d d' : D c} (ff : d -->[ id₁ c] d') : (local_iso_cleaving_1cell h (ff;;id_disp d') (idempunitor c)) ==>[ id₂ (id₁ c)] ff. Proof. assert (Heq : idempunitor c • runitor (identity c) = id2 (identity c)). { abstract (cbn ; rewrite <- lunitor_runitor_identity, linvunitor_lunitor ; reflexivity). } set (PP := disp_local_iso_cleaving_invertible_2cell h (ff;; id_disp d') (idempunitor c)). exact (transportf (λ x, _ ==>[x] _) Heq (PP •• disp_runitor ff)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat.