Definition total_set_category_pb_ump_1_mor_data : functor_data C₀ (total_setcategory_of_set_functor G₁). Proof. use make_functor_data. - refine (λ x, P₁ x ,, _). apply (αinv (P₁ x)). apply (# G₂ (pr1 (Hβ x))). exact (pr2 (P₂ x)). - refine (λ x y f, #P₁ f ,, _ · #(pr1 (αinv (P₁ y))) (# (pr1 (# G₂ (pr1 (Hβ y)))) (pr2 (# P₂ f)))). apply idtoiso. exact (total_set_category_pb_ump_1_mor_eq f). 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.