Lemma Signature_category_mor_diagram_pointwise (Hc : ∏ c, pr1 f1 c · pr1 f2 c = pr1 (α X) ((pr1 Y) c) · pr1 g2 c) : Signature_category_mor_diagram. Proof. apply nat_trans_eq_alt; intro c; simpl. unfold horcomp_data; simpl. rewrite functor_id, id_right; apply (Hc c). 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.whiskering. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.SIP. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SumOfSignatures.