Lemma unique_adjoint_equivalence_comp {B : bicat} (HB : is_univalent_2 B) {a b c : B} : ∏ (f : adjoint_equivalence a b) (g : adjoint_equivalence b c), comp_adjequiv f g = comp_adjoint_equivalence (pr1 HB) a b c f g. Proof. use (J_2_0 (pr1 HB) (λ a b f, _)). intros x g; simpl. unfold comp_adjoint_equivalence. rewrite J_2_0_comp. use subtypePath. { intro. exact (isaprop_left_adjoint_equivalence _ (pr2 HB)). } cbn. apply (isotoid_2_1 (pr2 HB)). use make_invertible_2cell. - exact (lunitor (pr1 g)). - is_iso. Qed.
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.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Section FromCompCartesian. End FromCompCartesian.