Lemma θ_Strength2_int_nicest_implies_θ_Strength2_int_nicer: θ_Strength2_int_nicest -> θ_Strength2_int_nicer. Proof. intro Hyp. intros X Z Z'. assert (HypX := Hyp X Z Z'). do 2 rewrite <- assoc. etrans. 2: { apply maponpaths. rewrite assoc. exact HypX. } apply pathsinv0. apply (id_left(a:=functor_compose (U Z ∙ U Z') (H X))). Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Notation.