Proposition associator_profunctor_nat_trans_mor_ob_comm {C₁ C₂ C₃ C₄ : category} (P₁ : C₁ ↛ C₂) (P₂ : C₂ ↛ C₃) (P₃ : C₃ ↛ C₄) {z : C₄} {w : C₁} {x : C₂} {y : C₃} (h : P₁ x w) (k : P₂ y x) (l : P₃ z y) : associator_profunctor_nat_trans_mor_ob P₁ P₂ P₃ z w x h (comp_profunctor_ob_in P₂ P₃ y k l) = comp_profunctor_ob_in (comp_profunctor P₁ P₂) P₃ y (comp_profunctor_ob_in P₁ P₂ x h k) l. Proof. unfold associator_profunctor_nat_trans_mor_ob. 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.FunctorCategory. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.opp_precat. 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.