Proposition prod_univalent_double_cat_pr2_lunitor : double_functor_lunitor (double_cat_double_lunitor (prod_univalent_double_cat D₁ D₂)) (double_cat_double_lunitor D₂) prod_univalent_double_cat_pr2_hor_id prod_univalent_double_cat_pr2_hor_comp. Proof. use make_double_functor_lunitor_double_cat. intros x y h ; cbn. rewrite <- square_assoc_v'. rewrite transportf_f_square. refine (!_). etrans. { do 2 apply maponpaths. apply (square_id_left_v (C := D₂)). } unfold transportb_square. rewrite transportf_square_postwhisker. rewrite transportf_f_square. rewrite <- id_h_square_id. etrans. { apply maponpaths. apply (lunitor_square (C := D₂)). } unfold transportb_square. rewrite transportf_f_square. etrans. { apply maponpaths. apply (square_id_right_v (C := D₂)). } unfold transportb_square. rewrite transportf_f_square. apply transportf_square_id. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.ProdOfTwosidedDispCat. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Limits.Products. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor.Basics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleTransformation. Require Import UniMath.Bicategories.DoubleCategories.Core.BicatOfDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.InvertiblesAndEquivalences. Require Import UniMath.Bicategories.DoubleCategories.DerivedLaws.TransportLaws. Require Import UniMath.Bicategories.DoubleCategories.DerivedLaws.Variations.