Definition preserves_subobject_classifier_disp_codomain_fiber_functor {C₁ C₂ : univalent_category} {T₁ : Terminal C₁} {T₂ : Terminal C₂} (P₁ : BinProducts C₁) (P₂ : BinProducts C₂) (Ω₁ : subobject_classifier T₁) (Ω₂ : subobject_classifier T₂) (F : C₁ ⟶ C₂) (HFT : preserves_terminal F) (HFP : preserves_binproduct F) (HFΩ : preserves_subobject_classifier F T₁ T₂ HFT) (x : C₁) : preserves_subobject_classifier (fiber_functor (disp_codomain_functor F) x) (codomain_fib_terminal x) (codomain_fib_terminal (F x)) (preserves_terminal_fiber_disp_codomain_functor F x). Proof. use preserves_chosen_to_preserves_subobject_classifier'. - apply is_univalent_cod_slice. - apply is_univalent_cod_slice. - exact (cod_fib_subobject_classifier T₁ P₁ Ω₁ x). - use preserves_chosen_to_preserves_chosen_subobject_classifier'. + apply is_univalent_cod_slice. + apply is_univalent_cod_slice. + exact (cod_fib_subobject_classifier T₂ P₂ Ω₂ (F x)). + use preserves_chosen_subobject_classifier_disp_codomain_fiber_functor. * exact HFT. * exact HFP. * exact HFΩ. 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).