Definition invertible_2cell_mor_between_preserves_cartesian {B : bicat} {e₁ e₂ b₁ b₂ : B} {p₁ : e₁ --> b₁} {p₂ : e₂ --> b₂} {fe fe' : e₁ --> e₂} (α : invertible_2cell fe fe') (H : mor_preserves_cartesian p₁ p₂ fe) : mor_preserves_cartesian p₁ p₂ fe'. Proof. intros x f g γ Hγ. assert (γ ▹ fe' • (_ ◃ α^-1) = (f ◃ α^-1) • (γ ▹ fe)) as p. { rewrite vcomp_whisker. apply idpath. } use (is_cartesian_2cell_sfib_postcomp _ _ _ _ p). - use invertible_is_cartesian_2cell_sfib. is_iso. - use vcomp_is_cartesian_2cell_sfib. + use invertible_is_cartesian_2cell_sfib. is_iso. + exact (H x f g γ Hγ). 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.DisplayedCats.StreetFibration. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence.