Definition conservative_invertible {B : bicat} {a b : B} {f₁ f₂ : a --> b} (β : f₁ ==> f₂) (Hβ : is_invertible_2cell β) (Hf₁ : conservative_1cell f₁) : conservative_1cell f₂. Proof. intros x g₁ g₂ α Hα. apply Hf₁. use eq_is_invertible_2cell. - exact ((g₁ ◃ β) • (α ▹ f₂) • (g₂ ◃ Hβ^-1)). - abstract (rewrite <- vcomp_whisker ; rewrite !vassocl ; rewrite lwhisker_vcomp ; rewrite vcomp_rinv ; rewrite lwhisker_id2 ; apply id2_right). - use is_invertible_2cell_vcomp. + use is_invertible_2cell_vcomp. * is_iso. * exact Hα. + is_iso. 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.