Definition cod_fib_subobject_classifier_unique_pb (χ : zg₂ --> pr_cod_fib P x Ω) (p : mp · χ = TerminalArrow (codomain_fib_terminal x) zg₁ · mor_to_pr_cod_fib P (codomain_fib_terminal x) (const_true x Ω)) (H : isPullback p) : isPullback (cod_fib_subobject_classifier_unique_eq χ p H). Proof. intros w h k r. use iscontraprop1. - exact (cod_fib_subobject_classifier_unique_pb_unique H h k). - simple refine (_ ,, _ ,, _). + exact (cod_fib_subobject_classifier_unique_pb_map H h k r). + apply cod_fib_subobject_classifier_unique_pb_pr1. + abstract (apply TerminalArrowEq). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.PullbackConstructions. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifier. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifierIso. Require Import UniMath.CategoryTheory.SubobjectClassifier.PreservesSubobjectClassifier. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseTerminal. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseSubobjectClassifier. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodMorphisms. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor. pose (BP := BinProductsFromPullbacks P T). pose (BP := BinProductsFromPullbacks P T).