Proposition is_local_property_parameterized_NNO : is_local_property parameterized_NNO_cat_property. Proof. use make_is_local_property. - intros C x N. exact (parameterized_NNO_prod_independent (C := slice_univ_cat_with_finlim x) _ (slice_parameterized_NNO N (pullbacks_univ_cat_with_finlim C) x)). - intros C N x y f ; cbn. use (preserves_parameterized_NNO_prod_independent (C₁ := slice_univ_cat_with_finlim y) (C₂ := slice_univ_cat_with_finlim x)). apply slice_parameterized_NNO_stable. - intros C₁ C₂ N₁ N₂ F HF x. use (preserves_parameterized_NNO_prod_independent (C₁ := slice_univ_cat_with_finlim x) (C₂ := slice_univ_cat_with_finlim (F x))). use codomain_functor_preserves_parameterized_NNO. + apply functor_finlim_preserves_terminal. + apply functor_finlim_preserves_binproduct. + 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.