Definition internal_sfib_cleaving_invertible : internal_sfib_cleaving f₂. Proof. intros x g₁ g₂ α. pose (ℓ := pr1 Hf₁ x g₁ g₂ (α • (g₂ ◃ Hβ^-1))). simple refine (_ ,, _ ,, _ ,, _ ,, _). - exact (pr1 ℓ). - exact (pr12 ℓ). - exact (comp_of_invertible_2cell (lwhisker_of_invertible_2cell _ (inv_of_invertible_2cell (β ,, Hβ))) (pr122 ℓ)). - apply (is_cartesian_2cell_invertible β Hβ). exact (pr1 (pr222 ℓ)). - abstract (simpl ; rewrite !vassocl ; use vcomp_move_L_pM ; [ is_iso | ] ; cbn ; rewrite <- vcomp_whisker ; use vcomp_move_R_Mp ; [ is_iso ; apply Hβ | ] ; cbn ; rewrite !vassocl ; exact (pr2 (pr222 ℓ))). 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.