Proposition strict_double_functor_hor_comp_mor {C₁ C₂ : strict_double_cat} (F : strict_double_functor C₁ C₂) {x₁ x₂ y₁ y₂ z₁ z₂ : C₁} {vx : x₁ -->v x₂} {vy : y₁ -->v y₂} {vz : z₁ -->v z₂} {h₁ : x₁ -->h y₁} {k₁ : y₁ -->h z₁} {h₂ : x₂ -->h y₂} {k₂ : y₂ -->h z₂} (s₁ : s_square vx vy h₁ h₂) (s₂ : s_square vy vz k₁ k₂) : #s F (s₁ ⋆h s₂) ⋆v id_h_to_s_square (strict_double_functor_hor_comp F h₂ k₂) = transportf_s_square (s_id_v_left _ @ !(s_id_v_right _)) (s_id_v_left _ @ !(s_id_v_right _)) (idpath _) (idpath _) (id_h_to_s_square (strict_double_functor_hor_comp F h₁ k₁) ⋆v (#s F s₁ ⋆h #s F s₂)). Proof. exact (is_natural_preserves_hor_comp (strict_double_functor_preserves_hor_comp F) s₁ s₂). Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.StrictTwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TransportLaws. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Basics.StrictDoubleCatBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.CatOfStrictDoubleCats.