Proposition cod_fib_subobject_classifier_unique : isaprop (∑ (χ : zg₂ --> pr_cod_fib P x Ω) (H : mp · χ = TerminalArrow (codomain_fib_terminal x) zg₁ · mor_to_pr_cod_fib P (codomain_fib_terminal x) (const_true x Ω)), isPullback H). Proof. use invproofirrelevance. intros χ₁ χ₂. use subtypePath. { intro. apply isaproptotal2. { intro. apply isaprop_isPullback. } intros. apply homset_property. } use eq_mor_cod_fib. use BinProductArrowsEq. - exact (mor_eq (pr1 χ₁) @ !(mor_eq (pr1 χ₂))). - use (subobject_classifier_map_eq Ω m). + exact (cod_fib_subobject_classifier_unique_eq (pr1 χ₁) (pr12 χ₁) (pr22 χ₁)). + exact (cod_fib_subobject_classifier_unique_eq (pr1 χ₂) (pr12 χ₂) (pr22 χ₂)). + apply cod_fib_subobject_classifier_unique_pb. + apply cod_fib_subobject_classifier_unique_pb. Qed.
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).