Lemma is_functor_functorial_composition_legacy_data (A B C : category) : is_functor (functorial_composition_legacy_data A B C). Proof. split. - red. intros FG. apply (nat_trans_eq C). intros x. apply remove_id_left. + apply idpath. + exact (functor_id (pr2 FG) ((pr1 (pr1 FG)) x)). - red. intros FG1 FG2 FG3 αβ1 αβ2. induction αβ1 as [α1 β1]. induction αβ2 as [α2 β2]. apply (nat_trans_eq C). intros a. simpl. unfold horcomp_data; simpl. rewrite <- ?assoc. apply cancel_precomposition. rewrite functor_comp. rewrite -> ?assoc. apply cancel_postcomposition. apply pathsinv0. apply nat_trans_ax. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.UnitorsAndAssociatorsForEndofunctors.