Definition Catlike_associator ( a b c d : category ) : nat_trans (functor_composite (pair_functor (functor_identity (functor_category a b)) (functorial_composition b c d )) (functorial_composition a b d )) (functor_composite (precategory_binproduct_assoc (functor_category a b ) (functor_category b c) (functor_category c d )) (functor_composite (pair_functor (functorial_composition a b c ) (functor_identity (functor_category c d))) (functorial_composition a c d ))). Proof. use tpair. - intros x. simpl. exists (λ x, identity _). intros oba oba' f. use (id_right _ @ !(id_left _)). - intros [F [G H]]. intros [F' [G' H']]. intros [f [g h]]. apply (nat_trans_eq_alt). intros oba. simpl. do 2 (unfold horcomp_data; simpl). rewrite id_right. rewrite id_left. rewrite (functor_comp H). rewrite assoc. apply idpath. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.Bicategories.WkCatEnrichment.prebicategory.