Lemma θ_Strength2_int_implies_θ_Strength2_int_nicer: θ_Strength2_int -> θ_Strength2_int_nicer. Proof. intro Hyp. intros X Z Z'. assert (HypX := Hyp X Z Z'). set (auxiso := functor_on_z_iso H (z_iso_inv (_,,(lassociator_CAT_pointwise_is_z_iso (U Z) (U Z') X)))). apply pathsinv0 in HypX. apply (z_iso_inv_on_left _ _ _ _ auxiso) in HypX. assumption. 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.