Proposition subobject_classifier_is_local_property : is_local_property subobject_classifier_cat_property. Proof. use make_is_local_property. - intros C x Ω. apply (codomain_fiberwise_subobject_classifier _ (pullbacks_univ_cat_with_finlim C) Ω). - intros C Ω x y f. apply (codomain_fiberwise_subobject_classifier _ (pullbacks_univ_cat_with_finlim C) Ω). - intros C₁ C₂ Ω₁ Ω₂ F HF x ; cbn. use preserves_subobject_classifier_disp_codomain_fiber_functor. + exact (terminal_univ_cat_with_finlim C₁). + exact (terminal_univ_cat_with_finlim C₂). + exact (binproducts_univ_cat_with_finlim C₁). + exact (binproducts_univ_cat_with_finlim C₂). + exact Ω₁. + exact Ω₂. + exact (functor_finlim_preserves_terminal F). + exact (functor_finlim_preserves_binproduct F). + exact HF. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.RegularAndExact.RegularEpi. Require Import UniMath.CategoryTheory.RegularAndExact.RegularEpiFacts. Require Import UniMath.CategoryTheory.RegularAndExact.RegularCategory. Require Import UniMath.CategoryTheory.RegularAndExact.ExactCategory. Require Import UniMath.CategoryTheory.Arithmetic.ParameterizedNNO. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. 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.CodFunctor. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodColimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.ColimitProperties. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodRegular. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodSubobjectClassifier. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodNNO. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.StrictInitial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.DisjointBinCoproducts. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifier. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifierIso. Require Import UniMath.CategoryTheory.SubobjectClassifier.PreservesSubobjectClassifier. Require Import UniMath.Bicategories.Core.Examples.StructuredCategories. Require Import UniMath.Bicategories.ComprehensionCat.LocalProperty.LocalProperties.