Proposition functor_double_id_eq {C₁ C₂ : category} {D₁ : twosided_disp_cat C₁ C₁} {D₂ : twosided_disp_cat C₂ C₂} {F : C₁ ⟶ C₂} {FF : twosided_disp_functor F F D₁ D₂} {I₁ : hor_id D₁} {I₂ : hor_id D₂} (IFF : double_functor_hor_id FF I₁ I₂) {x y : C₁} (f : x --> y) : double_id_mor I₂ (#F f) ;;2 functor_double_id_cell IFF y = transportb_disp_mor2 (id_right _ @ !(id_left _)) (id_right _ @ !(id_left _)) (functor_double_id_cell IFF x ;;2 #2 FF (double_id_mor I₁ f)). Proof. exact (pr2 IFF x y f). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. 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.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics.