Proposition comp_v_profunctor_square_mor_comm {C₁ C₂ D₁ D₂ E₁ E₂ : category} {F₁ : C₁ ⟶ C₂} {F₂ : D₁ ⟶ D₂} {F₃ : E₁ ⟶ E₂} {P₁ : C₁ ↛ D₁} {Q₁ : D₁ ↛ E₁} {P₂ : C₂ ↛ D₂} {Q₂ : D₂ ↛ E₂} (τ : profunctor_square F₂ F₁ P₁ P₂) (θ : profunctor_square F₃ F₂ Q₁ Q₂) (z : E₁) (x : C₁) (y : D₁) (h₁ : P₁ y x) (h₂ : Q₁ z y) : comp_v_profunctor_square_mor τ θ z x (comp_profunctor_ob_in _ _ y h₁ h₂) = comp_profunctor_ob_in P₂ Q₂ (F₂ y) (τ y x h₁) (θ z y h₂). Proof. unfold comp_v_profunctor_square_mor. rewrite from_comp_profunctor_ob_comm. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Categories.HSET.SetCoends. Require Import UniMath.CategoryTheory.Profunctors.Core. Require Import UniMath.CategoryTheory.Profunctors.Examples. Require Import UniMath.CategoryTheory.Profunctors.Transformation.