Definition left_adjoint_repr_to_left_adjoint_counit : invertible_2cell (l · r) (id₁ x). Proof. apply z_iso_to_inv2cell. refine (z_iso_comp (nat_z_iso_pointwise_z_iso (unit_nat_z_iso_from_adj_equivalence_of_cats (Hl x)) (l · r)) (z_iso_comp (functor_on_z_iso (right_adjoint (Hl x)) _) (z_iso_inv (nat_z_iso_pointwise_z_iso (unit_nat_z_iso_from_adj_equivalence_of_cats (Hl x)) (id₁ x))))). cbn. apply inv2cell_to_z_iso. exact (comp_of_invertible_2cell (rassociator_invertible_2cell _ _ _) (comp_of_invertible_2cell (lwhisker_of_invertible_2cell _ η) (comp_of_invertible_2cell (runitor_invertible_2cell _) (linvunitor_invertible_2cell _)))). 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.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions.