Definition total_set_category_pb_ump_1_mor_pr2_is_nat_z_iso : is_nat_z_iso total_set_category_pb_ump_1_mor_pr2. Proof. intro x. use is_z_iso_total_setcategory_of_set_functor. - exact (Hβ x). - exact (idtoiso (total_set_category_pb_ump_1_mor_pr2_is_nat_z_iso_eq x)). - abstract (cbn ; etrans ; [ apply maponpaths_2 ; refine (!_) ; apply (pr1_maponpaths_idtoiso (# G₂ (pr1 (Hβ x)))) | ] ; etrans ; [ refine (!_) ; apply (pr1_idtoiso_concat _ (total_set_category_pb_ump_1_mor_pr2_is_nat_z_iso_eq x)) | ] ; apply setcategory_eq_idtoiso). - abstract (cbn ; etrans ; [ apply maponpaths_2 ; refine (!_) ; apply (pr1_maponpaths_idtoiso (# G₂ (β x))) | ] ; etrans ; [ refine (!_) ; apply (pr1_idtoiso_concat _ (total_set_category_pb_ump_1_mor_pr2_eq x)) | ] ; apply setcategory_eq_idtoiso). 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.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.DisplayedCats.StreetOpFibration. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.GrothendieckConstruction.TotalCategory. Require Import UniMath.CategoryTheory.GrothendieckConstruction.Projection. Require Import UniMath.CategoryTheory.GrothendieckConstruction.IsosInTotal. Section PullbackFromTotal. End PullbackFromTotal.