proposition: Proposition disp_prebicat_ops_comp_cat : disp_prebicat_ops disp_prebicat_1_id_comp_cells_comp_cat.
proof: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq UniMath.CategoryTheory.DisplayedCats.ComprehensionC UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Preservation UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.FibTerminal UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispAdjunctions UniMath.Bicategories.DisplayedBicats.DispInvertibles UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat