Definition left_equivalence_invertible {B : bicat} {a b : B} {f g : a --> b} (g_equiv : left_equivalence g) {α : f ==> g} (Hα : is_invertible_2cell α) : left_equivalence f. Proof. simple refine ((_ ,, (_ ,, _)) ,, (_ ,, _)). - exact (left_adjoint_right_adjoint g_equiv). - exact ((left_adjoint_unit g_equiv) • ((Hα^-1) ▹ left_adjoint_right_adjoint g_equiv)). - exact ((left_adjoint_right_adjoint g_equiv ◃ α) • (left_adjoint_counit g_equiv)). - cbn. is_iso. apply g_equiv. - cbn. is_iso. apply g_equiv. 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.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.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Section InvertibleFromInternalSFib. End InvertibleFromInternalSFib.