Proposition disp_nat_trans_to_indexed_nat_trans_law : indexed_nat_trans_law disp_nat_trans_to_indexed_nat_trans_data. Proof. intros x y f xx ; cbn. refine (!_). use (cartesian_factorisation_unique (cartesian_disp_functor_on_cartesian G (HD₁ _ _ _ _))). rewrite mor_disp_transportf_postwhisker. rewrite assoc_disp_var. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite transport_f_f. refine (!_). rewrite mor_disp_transportf_postwhisker. rewrite assoc_disp_var. rewrite transport_f_f. etrans. { do 2 apply maponpaths. apply (disp_nat_trans_ax_var τ). } rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite assoc_disp. unfold transportb. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. apply maponpaths_2. apply homset_property. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedCategory. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedFunctor. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedTransformation. Require Import UniMath.CategoryTheory.IndexedCategories.FibrationToIndexedCategory. Require Import UniMath.CategoryTheory.IndexedCategories.CartesianToIndexedFunctor.