Definition invertible_2cell_between_preserves_cartesian {B : bicat} {e₁ e₂ b₁ b₂ : B} {p₁ p₁' : e₁ --> b₁} {p₂ p₂' : e₂ --> b₂} {fe fe' : e₁ --> e₂} (α : invertible_2cell p₁ p₁') (β : invertible_2cell p₂ p₂') (γ : invertible_2cell fe fe') (H : mor_preserves_cartesian p₁ p₂ fe) : mor_preserves_cartesian p₁' p₂' fe'. Proof. intros w h₁ h₂ ζ Hζ. use (is_cartesian_2cell_sfib_inv2cell β). use (invertible_2cell_mor_between_preserves_cartesian γ H). use (is_cartesian_2cell_sfib_inv2cell (inv_of_invertible_2cell α)). exact Hζ. 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.