(**

 The internal language of elementary toposes with a NNO

 In other files, we gave several extensions of the biequivalence between categories with
 finite limits and full DFL comprehension categories. Now we put these extensions together
 in order to obtain internal language theorems for several classes of toposes. The results
 are split over several files.

 In this file, we consider elementary toposes with an NNO. We approach these in the same
 way as ordinary toposes. The only difference being that we use another local property,
 which includes a parameterized natural numbers object. Since toposes are Cartesian closed,
 parameterized natural numbers objects are the same as ordinary ones.

 Contents
 1. The bicategory of elementary toposes with a natural numbers object
 2. Accessors for elementary toposes with a natural numbers object
 3. The bicategory of elementary topos comprehension categories with natural numbers
 4. Accessors for elementary toposes comprehension categories with natural numbers
 5. The biequivalence

                                                                                          *)
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Prelude.
Require Import UniMath.CategoryTheory.Arithmetic.ParameterizedNNO.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.Limits.Terminal.
Require Import UniMath.CategoryTheory.Limits.Pullbacks.
Require Import UniMath.CategoryTheory.Limits.Preservation.
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.LocallyCartesianClosed.LocallyCartesianClosed.
Require Import UniMath.CategoryTheory.LocallyCartesianClosed.Preservation.
Require Import UniMath.CategoryTheory.RegularAndExact.RegularCategory.
Require Import UniMath.CategoryTheory.RegularAndExact.RegularEpi.
Require Import UniMath.CategoryTheory.RegularAndExact.ExactCategory.
Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifier.
Require Import UniMath.CategoryTheory.SubobjectClassifier.PreservesSubobjectClassifier.
Require Import UniMath.Bicategories.Core.Bicat.
Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats.
Require Import UniMath.Bicategories.Core.Examples.StructuredCategories.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat.
Import DispBicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod.
Require Import UniMath.Bicategories.DisplayedBicats.DispBiequivalence.
Require Import UniMath.Bicategories.DisplayedBicats.DispPseudofunctor.
Require Import UniMath.Bicategories.DisplayedBicats.ProductDispBiequiv.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Require Import UniMath.Bicategories.PseudoFunctors.Biequivalence.
Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.
Require Import UniMath.Bicategories.ComprehensionCat.CompCatNotations.
Require Import UniMath.Bicategories.ComprehensionCat.DFLCompCat.
Require Import UniMath.Bicategories.ComprehensionCat.TypeFormers.PiTypes.
Require Import UniMath.Bicategories.ComprehensionCat.LocalProperty.LocalProperties.
Require Import UniMath.Bicategories.ComprehensionCat.LocalProperty.DFLCompCatWithProp.
Require Import UniMath.Bicategories.ComprehensionCat.LocalProperty.CatWithProp.
Require Import UniMath.Bicategories.ComprehensionCat.LocalProperty.Examples.
Require Import UniMath.Bicategories.ComprehensionCat.LocalProperty.DFLCompCatExamples.
Require Import UniMath.Bicategories.ComprehensionCat.Biequivalence.FinLimToDFLCompCat.
Require Import UniMath.Bicategories.ComprehensionCat.Biequivalence.DFLCompCatToFinLim.
Require Import UniMath.Bicategories.ComprehensionCat.Biequivalence.Unit.
Require Import UniMath.Bicategories.ComprehensionCat.Biequivalence.Counit.
Require Import UniMath.Bicategories.ComprehensionCat.Biequivalence.Biequiv.
Require Import UniMath.Bicategories.ComprehensionCat.Biequivalence.PiTypesBiequiv.
Require Import UniMath.Bicategories.ComprehensionCat.Biequivalence.LocalProperty.
Require Import UniMath.Bicategories.ComprehensionCat.InternalLanguageTopos.Pretopos.
Require Import UniMath.Bicategories.ComprehensionCat.InternalLanguageTopos.PiPretopos.

Local Open Scope cat.

(** * 1. The bicategory of elementary toposes with a natural numbers object *)
Definition disp_bicat_of_univ_topos_with_NNO
  : disp_bicat bicat_of_univ_cat_with_finlim
  := disp_dirprod_bicat
       (disp_bicat_of_univ_cat_with_cat_property topos_with_NNO_local_property)
       disp_bicat_univ_lccc.

Proposition disp_2cells_isaprop_disp_bicat_of_univ_topos_with_NNO
  : disp_2cells_isaprop disp_bicat_of_univ_topos_with_NNO.
Proof.
  use disp_2cells_isaprop_prod.
  - apply disp_2cells_isaprop_disp_bicat_of_univ_cat_with_cat_property.
  - exact disp_2cells_isaprop_univ_lccc.
Defined.

Proposition disp_locally_groupoid_disp_bicat_of_univ_topos_with_NNO
  : disp_locally_groupoid disp_bicat_of_univ_topos_with_NNO.
Proof.
  use disp_locally_groupoid_prod.
  - apply disp_locally_groupoid_disp_bicat_of_univ_cat_with_cat_property.
  - exact disp_locally_groupoid_univ_lccc.
Defined.

Proposition disp_univalent_2_disp_bicat_of_univ_topos_with_NNO
  : disp_univalent_2 disp_bicat_of_univ_topos_with_NNO.
Proof.
  use is_univalent_2_dirprod_bicat.
  - exact is_univalent_2_1_bicat_of_univ_cat_with_finlim.
  - apply disp_univalent_2_disp_bicat_univ_cat_with_cat_property.
  - exact disp_univalent_2_disp_bicat_univ_lccc.
Qed.

Proposition disp_univalent_2_0_disp_bicat_of_univ_topos_with_NNO
  : disp_univalent_2_0 disp_bicat_of_univ_topos_with_NNO.
Proof.
  apply disp_univalent_2_disp_bicat_of_univ_topos_with_NNO.
Qed.

Proposition disp_univalent_2_1_disp_bicat_of_univ_topos_with_NNO
  : disp_univalent_2_1 disp_bicat_of_univ_topos_with_NNO.
Proof.
  apply disp_univalent_2_disp_bicat_of_univ_topos_with_NNO.
Qed.

Definition bicat_of_univ_topos_with_NNO
  : bicat
  := total_bicat disp_bicat_of_univ_topos_with_NNO.

Proposition is_univalent_2_bicat_of_univ_topos_with_NNO
  : is_univalent_2 bicat_of_univ_topos_with_NNO.
Proof.
  use total_is_univalent_2.
  - exact disp_univalent_2_disp_bicat_of_univ_topos_with_NNO.
  - exact is_univalent_2_bicat_of_univ_cat_with_finlim.
Qed.

Proposition is_univalent_2_0_bicat_of_univ_topos_with_NNO
  : is_univalent_2_0 bicat_of_univ_topos_with_NNO.
Proof.
  apply is_univalent_2_bicat_of_univ_topos_with_NNO.
Qed.

Proposition is_univalent_2_1_bicat_of_univ_topos_with_NNO
  : is_univalent_2_1 bicat_of_univ_topos_with_NNO.
Proof.
  apply is_univalent_2_bicat_of_univ_topos_with_NNO.
Qed.

(** * 2. Accessors for elementary toposes with a natural numbers object *)
Definition univ_topos_with_NNO
  : UU
  := bicat_of_univ_topos_with_NNO.

Coercion univ_topos_with_NNO_to_univ_pretopos
         (E : univ_topos_with_NNO)
  : univ_pretopos.
Proof.
  use make_univ_pretopos.
  - exact (pr1 E).
  - exact (pr11 (pr111 (pr112 E))).
  - exact (pr121 (pr111 (pr112 E))).
  - exact (pr221 (pr111 (pr112 E))).
  - exact (pr2 (pr111 (pr112 E))).
  - exact (pr11 (pr211 (pr112 E))).
  - exact (pr21 (pr211 (pr112 E))).
  - exact (pr2 (pr211 (pr112 E))).
Defined.

Definition univ_topos_with_NNO_subobject_classifier
           (E : univ_topos_with_NNO)
  : subobject_classifier (terminal_univ_cat_with_finlim E)
  := pr21 (pr112 E).

Definition univ_topos_with_NNO_NNO
           (E : univ_topos_with_NNO)
  : parameterized_NNO
      (terminal_univ_cat_with_finlim E)
      (binproducts_univ_cat_with_finlim E)
  := pr2 (pr112 E).

Definition is_locally_cartesian_closed_univ_topos_with_NNO
           (E : univ_topos_with_NNO)
  : is_locally_cartesian_closed (pullbacks_univ_cat_with_finlim E)
  := pr122 E.

Definition make_univ_topos_with_NNO
           (E : univ_pi_pretopos)
           (Ω : subobject_classifier (terminal_univ_cat_with_finlim E))
           (N : parameterized_NNO
                  (terminal_univ_cat_with_finlim E)
                  (binproducts_univ_cat_with_finlim E))
  : univ_topos_with_NNO.
Proof.
  simple refine (_ ,, ((((((_ ,, (_ ,, _)) ,, _) ,, ((_ ,, _) ,, _)) ,, _) ,, _) ,, tt)
                   ,, _
                   ,, tt).
  - exact (pretopos_to_finlim E).
  - exact (strict_initial_univ_pretopos E).
  - exact (bincoproducts_univ_pretopos E).
  - exact (stable_bincoproducts_univ_pretopos E).
  - exact (disjoint_bincoproducts_univ_pretopos E).
  - exact (is_regular_category_coeqs_of_kernel_pair
             (is_regular_category_univ_pretopos E)).
  - exact (is_regular_category_regular_epi_pb_stable
             (is_regular_category_univ_pretopos E)).
  - exact (all_internal_eqrel_effective_univ_pretopos E).
  - exact Ω.
  - exact N.
  - exact (univ_pi_pretopos_lccc E).
Defined.

Definition functor_topos_with_NNO
           (E₁ E₂ : univ_topos_with_NNO)
  : UU
  := E₁ --> E₂.

Coercion functor_topos_with_NNO_to_functor_pretopos
         {E₁ E₂ : univ_topos_with_NNO}
         (F : functor_topos_with_NNO E₁ E₂)
  : functor_pretopos E₁ E₂.
Proof.
  use make_functor_pretopos.
  - exact (pr1 F).
  - exact (pr1 (pr111 (pr212 F))).
  - exact (pr2 (pr111 (pr212 F))).
  - exact (pr1 (pr211 (pr212 F))).
Defined.

Definition preserves_subobject_classifier_functor_topos_with_NNO
           {E₁ E₂ : univ_topos_with_NNO}
           (F : functor_topos_with_NNO E₁ E₂)
  : preserves_subobject_classifier
      F
      (terminal_univ_cat_with_finlim E₁)
      (terminal_univ_cat_with_finlim E₂)
      (functor_finlim_preserves_terminal F)
  := pr21 (pr212 F).

Definition preserves_NNO_functor_topos_with_NNO
           {E₁ E₂ : univ_topos_with_NNO}
           (F : functor_topos_with_NNO E₁ E₂)
  : preserves_parameterized_NNO
      (univ_topos_with_NNO_NNO E₁)
      (univ_topos_with_NNO_NNO E₂)
      F
      (functor_finlim_preserves_terminal F)
  := pr2 (pr212 F).

Definition preserves_locally_cartesian_closed_functor_topos_with_NNO
           {E₁ E₂ : univ_topos_with_NNO}
           (F : functor_topos_with_NNO E₁ E₂)
  : preserves_locally_cartesian_closed
      (functor_finlim_preserves_pullback F)
      (is_locally_cartesian_closed_univ_topos_with_NNO E₁)
      (is_locally_cartesian_closed_univ_topos_with_NNO E₂)
  := pr222 F.

Definition make_functor_topos_with_NNO
           {E₁ E₂ : univ_topos_with_NNO}
           (F : functor_pretopos E₁ E₂)
           (HF : preserves_subobject_classifier
                   F
                   (terminal_univ_cat_with_finlim E₁)
                   (terminal_univ_cat_with_finlim E₂)
                   (functor_finlim_preserves_terminal F))
           (HF' : preserves_locally_cartesian_closed
                    (functor_finlim_preserves_pullback F)
                    (is_locally_cartesian_closed_univ_topos_with_NNO E₁)
                    (is_locally_cartesian_closed_univ_topos_with_NNO E₂))
           (HF'' : preserves_parameterized_NNO
                     (univ_topos_with_NNO_NNO E₁)
                     (univ_topos_with_NNO_NNO E₂)
                     F
                     (functor_finlim_preserves_terminal F))
  : functor_topos_with_NNO E₁ E₂.
Proof.
  simple refine (_ ,, (tt ,, (((_ ,, _) ,, _ ,, tt) ,, _) ,, _) ,, tt ,, _).
  - exact (functor_pretopos_to_functor_finlim F).
  - exact (preserves_initial_functor_pretopos F).
  - exact (preserves_bincoproduct_functor_pretopos F).
  - exact (preserves_regular_epi_functor_pretopos F).
  - exact HF.
  - exact HF''.
  - exact HF'.
Defined.

Definition nat_trans_topos_with_NNO
           {E₁ E₂ : univ_topos_with_NNO}
           (F G : functor_topos_with_NNO E₁ E₂)
  : UU
  := F ==> G.

Coercion nat_trans_topos_with_NNO_to_nat_trans_finlim
         {E₁ E₂ : univ_topos_with_NNO}
         {F G : functor_topos_with_NNO E₁ E₂}
         (τ : nat_trans_topos_with_NNO F G)
  : nat_trans_finlim F G
  := pr1 τ.

Definition make_nat_trans_topos_with_NNO
           {E₁ E₂ : univ_topos_with_NNO}
           {F G : functor_topos_with_NNO E₁ E₂}
           (τ : F ⟹ G)
  : nat_trans_topos_with_NNO F G
  := make_nat_trans_finlim τ ,, (tt ,, tt) ,, (tt ,, tt).

Proposition nat_trans_topos_with_NNO_eq
            {E₁ E₂ : univ_topos_with_NNO}
            {F G : functor_topos_with_NNO E₁ E₂}
            {τ₁ τ₂ : nat_trans_topos_with_NNO F G}
            (p : ∏ (x : E₁), τ₁ x = τ₂ x)
  : τ₁ = τ₂.
Proof.
  use subtypePath.
  {
    intro.
    use isaproptotal2.
    - intro.
      use isapropdirprod ; apply isapropunit.
    - intros.
      use pathsdirprod ; apply isapropunit.
  }
  use nat_trans_finlim_eq.
  exact p.
Qed.

(** * 3. The bicategory of elementary topos comprehension categories with natural numbers *)
Definition disp_bicat_univ_topos_with_NNO_language
  : disp_bicat bicat_of_dfl_full_comp_cat
  := disp_dirprod_bicat
       (disp_bicat_of_cat_property_dfl_full_comp_cat topos_with_NNO_local_property)
       disp_bicat_of_pi_type_dfl_full_comp_cat.

Proposition disp_2cells_isaprop_disp_bicat_univ_topos_with_NNO_language
  : disp_2cells_isaprop disp_bicat_univ_topos_with_NNO_language.
Proof.
  use disp_2cells_isaprop_prod.
  - apply disp_2cells_isaprop_disp_bicat_of_cat_property_dfl_full_comp_cat.
  - exact disp_2cells_isaprop_disp_bicat_of_pi_type_dfl_full_comp_cat.
Defined.

Proposition disp_locally_groupoid_disp_bicat_univ_topos_with_NNO_language
  : disp_locally_groupoid disp_bicat_univ_topos_with_NNO_language.
Proof.
  use disp_locally_groupoid_prod.
  - apply disp_locally_groupoid_disp_bicat_of_cat_property_dfl_full_comp_cat.
  - exact disp_locally_groupoid_disp_bicat_of_pi_type_dfl_full_comp_cat.
Defined.

Proposition disp_univalent_2_disp_bicat_univ_topos_with_NNO_language
  : disp_univalent_2 disp_bicat_univ_topos_with_NNO_language.
Proof.
  use is_univalent_2_dirprod_bicat.
  - exact is_univalent_2_1_bicat_of_dfl_full_comp_cat.
  - apply univalent_2_disp_bicat_of_cat_property_dfl_full_comp_cat.
  - exact univalent_2_disp_bicat_of_pi_type_dfl_full_comp_cat.
Qed.

Proposition disp_univalent_2_0_disp_bicat_univ_topos_with_NNO_language
  : disp_univalent_2_0 disp_bicat_univ_topos_with_NNO_language.
Proof.
  apply disp_univalent_2_disp_bicat_univ_topos_with_NNO_language.
Qed.

Proposition disp_univalent_2_1_disp_bicat_univ_topos_with_NNO_language
  : disp_univalent_2_1 disp_bicat_univ_topos_with_NNO_language.
Proof.
  apply disp_univalent_2_disp_bicat_univ_topos_with_NNO_language.
Qed.

Definition univ_topos_with_NNO_language
  : bicat
  := total_bicat disp_bicat_univ_topos_with_NNO_language.

Proposition is_univalent_2_univ_topos_with_NNO_language
  : is_univalent_2 univ_topos_with_NNO_language.
Proof.
  use total_is_univalent_2.
  - exact disp_univalent_2_disp_bicat_univ_topos_with_NNO_language.
  - exact is_univalent_2_bicat_of_dfl_full_comp_cat.
Qed.

Proposition is_univalent_2_0_univ_topos_with_NNO_language
  : is_univalent_2_0 univ_topos_with_NNO_language.
Proof.
  apply is_univalent_2_univ_topos_with_NNO_language.
Qed.

Proposition is_univalent_2_1_univ_topos_with_NNO_language
  : is_univalent_2_1 univ_topos_with_NNO_language.
Proof.
  apply is_univalent_2_univ_topos_with_NNO_language.
Qed.

(** * 4. Accessors for elementary toposes comprehension categories with natural numbers *)
Definition topos_NNO_comp_cat
  : UU
  := univ_topos_with_NNO_language.

Coercion topos_NNO_comp_cat_to_dfl_comp_cat
         (C : topos_NNO_comp_cat)
  : dfl_full_comp_cat
  := pr1 C.

Definition fiberwise_strict_initial_topos_NNO_comp_cat
           (C : topos_NNO_comp_cat)
  : fiberwise_cat_property
      strict_initial_local_property
      (topos_NNO_comp_cat_to_dfl_comp_cat C)
  := local_property_conj_left
       (local_property_sub_left
          (local_property_conj_left
             (local_property_conj_left
                (local_property_conj_left
                   (pr112 C))))).

Definition fiberwise_bincoproduct_topos_NNO_comp_cat
           (C : topos_NNO_comp_cat)
  : fiberwise_cat_property
      stable_bincoproducts_local_property
      (topos_NNO_comp_cat_to_dfl_comp_cat C)
  := local_property_conj_right
       (local_property_sub_left
          (local_property_conj_left
             (local_property_conj_left
                (local_property_conj_left
                   (pr112 C))))).

Definition fiberwise_regular_topos_NNO_comp_cat
           (C : topos_NNO_comp_cat)
  : fiberwise_cat_property
      regular_local_property
      (topos_NNO_comp_cat_to_dfl_comp_cat C)
  := local_property_conj_left
       (local_property_conj_right
          (local_property_conj_left
             (local_property_conj_left (pr112 C)))).

Definition fiberwise_effective_eqrel_topos_NNO_comp_cat
           (C : topos_NNO_comp_cat)
  : fiberwise_cat_property
      all_eqrel_effective_local_property
      (topos_NNO_comp_cat_to_dfl_comp_cat C)
  := local_property_conj_right
       (local_property_conj_right
          (local_property_conj_left
             (local_property_conj_left (pr112 C)))).

Definition fiberwise_subobject_classifier_topos_NNO_comp_cat
           (C : topos_NNO_comp_cat)
  : fiberwise_cat_property
      subobject_classifier_local_property
      (topos_NNO_comp_cat_to_dfl_comp_cat C)
  := local_property_conj_right
       (local_property_conj_left (pr112 C)).

Definition fiberwise_nno_topos_NNO_comp_cat
           (C : topos_NNO_comp_cat)
  : fiberwise_cat_property
      parameterized_NNO_local_property
      (topos_NNO_comp_cat_to_dfl_comp_cat C)
  := local_property_conj_right (pr112 C).

Definition comp_cat_dependent_prod_topos_NNO_comp_cat
           (C : topos_NNO_comp_cat)
  : comp_cat_dependent_prod C
  := pr122 C.

Definition topos_NNO_comp_cat_functor
           (C₁ C₂ : topos_NNO_comp_cat)
  : UU
  := C₁ --> C₂.

Coercion topos_NNO_comp_cat_functor_to_dfl_comp_cat_functor
         {C₁ C₂ : topos_NNO_comp_cat}
         (F : topos_NNO_comp_cat_functor C₁ C₂)
  : dfl_full_comp_cat_functor
      (topos_NNO_comp_cat_to_dfl_comp_cat C₁)
      (topos_NNO_comp_cat_to_dfl_comp_cat C₂)
  := pr1 F.

Definition fiberwise_strict_initial_topos_NNO_comp_cat_functor
           {C₁ C₂ : topos_NNO_comp_cat}
           (F : topos_NNO_comp_cat_functor C₁ C₂)
  : fiberwise_cat_property_functor
      (topos_NNO_comp_cat_functor_to_dfl_comp_cat_functor F)
      (fiberwise_strict_initial_topos_NNO_comp_cat C₁)
      (fiberwise_strict_initial_topos_NNO_comp_cat C₂)
  := local_property_functor_conj_left
       (local_property_functor_sub_left
          (local_property_functor_conj_left
             (local_property_functor_conj_left
                (local_property_functor_conj_left (pr212 F))))).

Definition fiberwise_bincoproduct_topos_NNO_comp_cat_functor
           {C₁ C₂ : topos_NNO_comp_cat}
           (F : topos_NNO_comp_cat_functor C₁ C₂)
  : fiberwise_cat_property_functor
      (topos_NNO_comp_cat_functor_to_dfl_comp_cat_functor F)
      (fiberwise_bincoproduct_topos_NNO_comp_cat C₁)
      (fiberwise_bincoproduct_topos_NNO_comp_cat C₂)
  := local_property_functor_conj_right
       (local_property_functor_sub_left
          (local_property_functor_conj_left
             (local_property_functor_conj_left
                (local_property_functor_conj_left (pr212 F))))).

Definition fiberwise_regular_topos_NNO_comp_cat_functor
           {C₁ C₂ : topos_NNO_comp_cat}
           (F : topos_NNO_comp_cat_functor C₁ C₂)
  : fiberwise_cat_property_functor
      (topos_NNO_comp_cat_functor_to_dfl_comp_cat_functor F)
      (fiberwise_regular_topos_NNO_comp_cat C₁)
      (fiberwise_regular_topos_NNO_comp_cat C₂)
  := local_property_functor_conj_left
       (local_property_functor_conj_right
          (local_property_functor_conj_left
             (local_property_functor_conj_left (pr212 F)))).

Definition fiberwise_subobject_classifier_topos_NNO_comp_cat_functor
           {C₁ C₂ : topos_NNO_comp_cat}
           (F : topos_NNO_comp_cat_functor C₁ C₂)
  : fiberwise_cat_property_functor
      (topos_NNO_comp_cat_functor_to_dfl_comp_cat_functor F)
      (fiberwise_subobject_classifier_topos_NNO_comp_cat C₁)
      (fiberwise_subobject_classifier_topos_NNO_comp_cat C₂)
  := local_property_functor_conj_right
       (local_property_functor_conj_left (pr212 F)).

Definition fiberwise_nno_topos_NNO_comp_cat_functor
           {C₁ C₂ : topos_NNO_comp_cat}
           (F : topos_NNO_comp_cat_functor C₁ C₂)
  : fiberwise_cat_property_functor
      (topos_NNO_comp_cat_functor_to_dfl_comp_cat_functor F)
      (fiberwise_nno_topos_NNO_comp_cat C₁)
      (fiberwise_nno_topos_NNO_comp_cat C₂)
  := local_property_functor_conj_right (pr212 F).

Definition preserves_comp_cat_dependent_prod_topos_NNO_comp_cat_functor
           {C₁ C₂ : topos_NNO_comp_cat}
           (F : topos_NNO_comp_cat_functor C₁ C₂)
  : preserves_comp_cat_dependent_prod
      (topos_NNO_comp_cat_functor_to_dfl_comp_cat_functor F)
      (comp_cat_dependent_prod_topos_NNO_comp_cat C₁)
      (comp_cat_dependent_prod_topos_NNO_comp_cat C₂)
  := pr222 F.

Definition topos_NNO_comp_cat_nat_trans
           {C₁ C₂ : topos_NNO_comp_cat}
           (F G : topos_NNO_comp_cat_functor C₁ C₂)
  : UU
  := F ==> G.

Coercion topos_NNO_comp_cat_nat_trans_to_dfl_full_comp_cat_nat_trans
         {C₁ C₂ : topos_NNO_comp_cat}
         {F G : topos_NNO_comp_cat_functor C₁ C₂}
         (τ : topos_NNO_comp_cat_nat_trans F G)
  : dfl_full_comp_cat_nat_trans
      (topos_NNO_comp_cat_functor_to_dfl_comp_cat_functor F)
      (topos_NNO_comp_cat_functor_to_dfl_comp_cat_functor G)
  := pr1 τ.

(** * 5. The biequivalence *)
Definition disp_psfunctor_lang_univ_topos_with_NNO
  : disp_psfunctor
      disp_bicat_of_univ_topos_with_NNO
      disp_bicat_univ_topos_with_NNO_language
      finlim_biequiv_dfl_comp_cat_psfunctor.
Proof.
  refine (prod_disp_psfunctor
            _ _ _ _
            (finlim_biequiv_dfl_comp_cat_disp_psfunctor_local_property
               topos_with_NNO_local_property)
            finlim_biequiv_dfl_comp_cat_disp_psfunctor_pi_types).
  - apply disp_2cells_isaprop_disp_bicat_of_cat_property_dfl_full_comp_cat.
  - apply disp_locally_groupoid_disp_bicat_of_cat_property_dfl_full_comp_cat.
  - exact disp_2cells_isaprop_disp_bicat_of_pi_type_dfl_full_comp_cat.
  - exact disp_locally_groupoid_disp_bicat_of_pi_type_dfl_full_comp_cat.
Defined.

Definition lang_univ_topos_with_NNO
  : psfunctor
      bicat_of_univ_topos_with_NNO
      univ_topos_with_NNO_language
  := total_psfunctor
       _ _ _
       disp_psfunctor_lang_univ_topos_with_NNO.

Definition internal_language_univ_topos_with_NNO
  : is_biequivalence lang_univ_topos_with_NNO.
Proof.
  refine (total_is_biequivalence
            _ _ _
            (prod_disp_is_biequivalence_data
               _ _ _ _
               _ _ _ _
               _ _ _ _
               _ _ _ _
               (finlim_biequiv_dfl_comp_cat_psfunctor_local_property
                  topos_with_NNO_local_property)
               finlim_biequiv_dfl_comp_cat_psfunctor_pi_types)).
  - apply disp_2cells_isaprop_disp_bicat_of_univ_cat_with_cat_property.
  - apply disp_locally_groupoid_disp_bicat_of_univ_cat_with_cat_property.
  - exact disp_2cells_isaprop_univ_lccc.
  - exact disp_locally_groupoid_univ_lccc.
Defined.
