Definition pre_comp_post_comp_commute_z_iso {b₁ b₂ : B} (f : b₁ --> b₂) {c₁ c₂ : B} (g : c₁ --> c₂) : nat_z_iso (pre_comp c₁ f ∙ post_comp b₁ g) (post_comp b₂ g ∙ pre_comp c₂ f). Proof. use make_nat_z_iso. - exact (pre_comp_post_comp_commute f g). - intro. use is_inv2cell_to_is_z_iso ; cbn. is_iso. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.IsoCommaCategory. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Properties.ClosedUnderInvertibles. Require Import UniMath.Bicategories.Limits.Pullbacks. Require Import UniMath.Bicategories.Limits.PullbackFunctions. Import PullbackFunctions.Notations. Require Import UniMath.Bicategories.Limits.PullbackEquivalences. Require Import UniMath.Bicategories.Limits.Examples.BicatOfUnivCatsLimits.