Proposition eq_hcomp_square {C : double_cat} {x₁ x₂ y₁ y₂ z₁ z₂ : C} {v₁ : x₁ -->v x₂} {v₂ v₂' : y₁ -->v y₂} {v₃ : z₁ -->v z₂} {h₁ : x₁ -->h y₁} {h₂ : y₁ -->h z₁} {k₁ : x₂ -->h y₂} {k₂ : y₂ -->h z₂} {s₁ : square v₁ v₂ h₁ k₁} {s₁' : square v₁ v₂' h₁ k₁} {s₂ : square v₂ v₃ h₂ k₂} {s₂' : square v₂' v₃ h₂ k₂} (p : v₂ = v₂') (q₁ : s₁ = transportf_square (idpath _) (!p) s₁') (q₂ : s₂ = transportf_square (!p) (idpath _) s₂') : s₁ ⋆h s₂ = s₁' ⋆h s₂'. Proof. induction p. cbn in q₁, q₂. induction q₁, q₂. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats.