Definition is_cartesian_2cell_sfib_pointwise_cartesian (x : pr1 C₀) : is_cartesian_sfib F (pr1 α x). Proof. pose (i := invertible_between_cartesians Hα pointwise_lift_nat_trans_is_cartesian_2cell is_cartesian_2cell_sfib_pointwise_cartesian_inv2cell is_cartesian_2cell_sfib_pointwise_cartesian_eq). use is_cartesian_sfib_eq. - exact (nat_z_iso_pointwise_z_iso (invertible_2cell_to_nat_z_iso _ _ i) x · pr1 pointwise_lift_nat_trans x). - exact (cartesian_factorization_sfib_commute _ _ _ _ _). - use comp_is_cartesian_sfib. + apply z_iso_is_cartesian_sfib. apply z_iso_is_z_isomorphism. + exact (pr222 (HF (pr1 G₂ x) (pr1 F (pr1 G₁ x)) (# (pr1 F) (pr1 α x)))). 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.