Definition street_fib_is_internal_sfib_cleaving_lift_data : functor_data (pr1 C₀) (pr1 C₁). Proof. use make_functor_data. - exact (λ x, pr1 (HF (pr1 G₂ x) (pr1 G₁ x) (pr1 α x))). - intros x y f ; cbn. use (cartesian_factorization_sfib _ (pr222 (HF (pr1 G₂ y) (pr1 G₁ y) (pr1 α y)))). + exact (pr112 (HF (pr1 G₂ x) (pr1 G₁ x) (pr1 α x)) · # (pr1 G₂) f). + exact (pr212 (HF (pr1 G₂ x) (pr1 G₁ x) (pr1 α x)) · # (pr1 G₁) f · inv_from_z_iso (pr212 (HF (pr1 G₂ y) (pr1 G₁ y) (pr1 α y)))). + abstract (rewrite functor_comp ; rewrite (pr122 (HF (pr1 G₂ x) (pr1 G₁ x) (pr1 α x))) ; rewrite !assoc' ; apply maponpaths ; refine (!(nat_trans_ax α _ _ f) @ _) ; apply maponpaths ; refine (!_) ; use z_iso_inv_on_right ; rewrite (pr122 (HF (pr1 G₂ y) (pr1 G₁ y) (pr1 α y))) ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.DisplayedCats.StreetFibration. Require Import UniMath.CategoryTheory.DisplayedCats.StreetOpFibration. 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.Univalence. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration.