Proposition profunctor_comp {C₁ C₂ : category} (P : C₁ ↛ C₂) {y₁ y₂ y₃: C₂} (g₂ : y₃ --> y₂) (g₁ : y₂ --> y₁) {x₁ x₂ x₃ : C₁} (f₁ : x₁ --> x₂) (f₂ : x₂ --> x₃) (h : P y₁ x₁) : P #[ g₂ · g₁ , f₁ · f₂ ] h = P #[ g₂ , f₂ ] (P #[ g₁ , f₁ ] h). Proof. exact (eqtohomot (@functor_comp _ _ P (_ ,, _) (_ ,, _) (_ ,, _) (_ ,, _) (_ ,, _)) h). Qed.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Categories.HSET.All.