Definition from_pb_cartesian : is_cartesian_2cell_sfib p₂ (α ▹ fe). Proof. assert (p : δ • β = α). { apply is_cartesian_2cell_sfib_factor_comm. } use (is_cartesian_eq _ (maponpaths (λ z, z ▹ fe) p)). use (is_cartesian_eq _ (rwhisker_vcomp _ _ _)). use vcomp_is_cartesian_2cell_sfib. - apply invertible_is_cartesian_2cell_sfib. is_iso. apply property_from_invertible_2cell. - unfold β, pb_of_sfib_cleaving_cell. rewrite (pb_ump_cell_pr1 H). use vcomp_is_cartesian_2cell_sfib. + apply invertible_is_cartesian_2cell_sfib. apply property_from_invertible_2cell. + apply internal_sfib_cleaving_is_cartesian. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Examples.OpCellBicat. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Morphisms.Properties.Composition. Require Import UniMath.Bicategories.Limits.Pullbacks. Require Import UniMath.Bicategories.Limits.Examples.OpCellBicatLimits. Section FromPBCartesian. End FromPBCartesian.