Definition associator_profunctor_nat_trans_mor_ob {C₁ C₂ C₃ C₄ : category} (P₁ : C₁ ↛ C₂) (P₂ : C₂ ↛ C₃) (P₃ : C₃ ↛ C₄) (z : C₄) (w : C₁) (x : C₂) (h : P₁ x w) : comp_profunctor P₂ P₃ z x → comp_profunctor (comp_profunctor P₁ P₂) P₃ z w. Proof. use from_comp_profunctor_ob. - refine (λ y h' h'', comp_profunctor_ob_in _ _ y _ h''). exact (comp_profunctor_ob_in P₁ P₂ x h h'). - abstract (intros y₁ y₂ g k k' ; cbn ; rewrite comp_profunctor_ob_in_comm ; cbn ; rewrite comp_profunctor_mor_comm ; rewrite rmap_id ; apply idpath). Defined.
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.