Proposition disp_prebicat_ops_comp_cat : disp_prebicat_ops disp_prebicat_1_id_comp_cells_comp_cat. Proof. repeat split. - intros C₁ C₂ F χ₁ χ₂ Fχ x xx ; cbn. rewrite id_left. etrans. { apply maponpaths. apply comprehension_functor_mor_id. } apply id_right. - intros C₁ C₂ F χ₁ χ₂ Fχ x xx ; cbn. rewrite functor_id. rewrite !id_left. etrans. { apply maponpaths. apply comprehension_functor_mor_id. } apply id_right. - intros C₁ C₂ F χ₁ χ₂ Fχ x xx ; cbn. rewrite id_left, id_right. etrans. { apply maponpaths. apply comprehension_functor_mor_id. } apply id_right. - intros C₁ C₂ F χ₁ χ₂ Fχ x xx ; cbn. rewrite functor_id. rewrite !id_left. etrans. { apply maponpaths. apply comprehension_functor_mor_id. } apply id_right. - intros C₁ C₂ F χ₁ χ₂ Fχ x xx ; cbn. rewrite id_left, id_right. etrans. { apply maponpaths. apply comprehension_functor_mor_id. } apply id_right. - intros C₁ C₂ C₃ C₄ F₁ F₂ F₃ χ₁ χ₂ χ₃ χ₄ Fχ₁ Fχ₂ Fχ₃ x xx ; cbn. rewrite id_left. etrans. { apply maponpaths. apply comprehension_functor_mor_id. } rewrite id_right. rewrite functor_comp. rewrite !assoc'. apply idpath. - intros C₁ C₂ C₃ C₄ F₁ F₂ F₃ χ₁ χ₂ χ₃ χ₄ Fχ₁ Fχ₂ Fχ₃ x xx ; cbn. rewrite id_left. etrans. { apply maponpaths. apply comprehension_functor_mor_id. } rewrite id_right. rewrite functor_comp. rewrite !assoc'. apply idpath. - intros C₁ C₂ F₁ F₂ F₃ τ₁ τ₂ χ₁ χ₂ χ₃ Fχ₁ Fχ₂ p q x xx ; cbn ; cbn in p, q. etrans. { apply maponpaths. apply comprehension_functor_mor_comp. } rewrite !assoc. etrans. { apply maponpaths_2. apply p. } rewrite !assoc'. etrans. { apply maponpaths. apply q. } apply idpath. - intros C₁ C₂ C₃ F G₁ G₂ τ χ₁ χ₂ χ₃ Fχ Gχ₁ Gχ₂ p x xx ; cbn ; cbn in p. rewrite !assoc'. etrans. { apply maponpaths. apply p. } rewrite !assoc. apply maponpaths_2. apply nat_trans_ax. - intros C₁ C₂ C₃ F₁ F₂ G τ χ₁ χ₂ χ₃ Fχ₁ Fχ₂ Gχ p x xx ; cbn ; cbn in p. rewrite !assoc'. etrans. { apply maponpaths. refine (!_). apply comprehension_nat_trans_comm. } rewrite !assoc. apply maponpaths_2. rewrite <- !functor_comp. apply maponpaths. apply p. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.ComprehensionC. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.FibTerminal. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat.