Definition double_nat_trans_hor_id {C₁ C₂ : category} {D₁ : twosided_disp_cat C₁ C₁} {D₂ : twosided_disp_cat C₂ C₂} {F G : C₁ ⟶ C₂} {FF : twosided_disp_functor F F D₁ D₂} {GG : twosided_disp_functor G G D₁ D₂} {τ : F ⟹ G} (ττ : twosided_disp_nat_trans τ τ FF GG) {I₁ : hor_id D₁} {I₂ : hor_id D₂} (FI : double_functor_hor_id FF I₁ I₂) (GI : double_functor_hor_id GG I₁ I₂) : UU := ∏ (x : C₁), functor_double_id_cell FI x ;;2 ττ x x (double_id I₁ x) = transportb_disp_mor2 (id_left _ @ !(id_right _)) (id_left _ @ !(id_right _)) (double_id_mor I₂ (τ x) ;;2 functor_double_id_cell GI x).
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.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor.