Lemma rassociator_fun_natural {a b c d : C} : is_nat_trans (precategory_binproduct_assoc (hom_precategory a b) (hom_precategory b c) (hom_precategory c d) ∙ pair_functor hcomp_functor (functor_identity _) ∙ hcomp_functor) (pair_functor (functor_identity _) hcomp_functor ∙ hcomp_functor) rassociator_fun. Proof. red; cbn. intros (f1, (f2, f3)) (g1, (g2, g3)). unfold precategory_binproduct_mor, hom_ob_mor. cbn. unfold precategory_binproduct_mor, hom_ob_mor. cbn. intros (x1, (x2, x3)). cbn. unfold rassociator_fun. cbn. apply hcomp_rassoc. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.PrecategoryBinProduct.