Lemma ProductStrength2' : θ_Strength2_int θ. Proof. clear S11 S12 S21 S22 S11' S21'; intros X Z Z'. apply nat_trans_eq_alt; intro x; simpl. rewrite id_left. eapply pathscomp0; [apply BinProductOfArrows_comp|]. apply pathsinv0. eapply pathscomp0; [apply BinProductOfArrows_comp|]. apply pathsinv0, maponpaths_12. - assert (Ha_x := nat_trans_eq_pointwise (S12' X Z Z') x). simpl in Ha_x; rewrite id_left in Ha_x. exact Ha_x. - assert (Ha_x := nat_trans_eq_pointwise (S22' X Z Z') x). simpl in Ha_x; rewrite id_left in Ha_x. exact Ha_x. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Chains.Chains. Require Import UniMath.CategoryTheory.Chains.OmegaCocontFunctors. Require Import UniMath.CategoryTheory.exponentials.