Definition preserves_subobject_classifier_cod_pb {C : univalent_category} (T : Terminal C) (P : Pullbacks C) (Ω : subobject_classifier T) {x y : C} (f : x --> y) : preserves_subobject_classifier (cod_pb P f) (codomain_fib_terminal y) (codomain_fib_terminal x) (codomain_fib_preserves_terminal P f). Proof. pose (BP := BinProductsFromPullbacks P T). use preserves_chosen_to_preserves_subobject_classifier'. - apply is_univalent_cod_slice. - apply is_univalent_cod_slice. - exact (cod_fib_subobject_classifier T BP Ω y). - use preserves_chosen_to_preserves_chosen_subobject_classifier'. + apply is_univalent_cod_slice. + apply is_univalent_cod_slice. + exact (cod_fib_subobject_classifier T BP Ω x). + apply cod_pb_preserves_chosen_subobject_classifier. 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).