Proposition cod_pb_preserves_chosen_subobject_classifier_eq : # (cod_pb P f) (cod_fib_subobject_classifier T BP Ω y) · cod_pb_preserves_chosen_subobject_classifier_z_iso = TerminalArrow (codomain_fib_terminal x) (cod_pb P f (codomain_fib_terminal y)) · cod_fib_subobject_classifier T BP Ω x. Proof. use eq_mor_cod_fib. rewrite !comp_in_cod_fib. etrans. { apply maponpaths_2. apply maponpaths. apply cod_fiber_functor_on_mor. } cbn. unfold map_from_pb_prod. use BinProductArrowsEq. - rewrite !assoc'. rewrite !BinProductPr1Commutes. rewrite PullbackArrow_PullbackPr2. rewrite !id_right. apply idpath. - rewrite !assoc'. rewrite BinProductPr2Commutes. rewrite !assoc. rewrite PullbackArrow_PullbackPr1. rewrite !assoc'. rewrite !BinProductPr2Commutes. rewrite id_left. unfold const_true. rewrite !assoc. apply maponpaths_2. apply TerminalArrowEq. 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).