Lemma total_sigma_cell_eq {a b : total_bicat E} {f g : total_bicat E ⟦a,b⟧} (x y : f ==> g) (eq1 : pr1 x = pr1 y) (eq2 : pr2 x = transportb (λ z, pr2 f ==>[z] pr2 g) eq1 (pr2 y)) : x = y. Proof. induction x as (x, xx). induction y as (y, yy). cbn in *. induction eq1. cbn in *. apply pair_path_in2. exact eq2. 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.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.