Lemma θ_nat_2_pointwise (X : [C, D']) (Z Z' : Ptd) (f : Z --> Z') (c : C) : # (pr1 (H X)) ((pr1 f) c) · pr1 (θ (X ⊗ Z')) c = pr1 (θ (X ⊗ Z)) c · pr1 (# H (identity X ⋆ pr1 f)) c . Proof. set (t := θ_nat_2 X _ _ f). set (t' := nat_trans_eq_weq (homset_property D) _ _ t c). clearbody t'; clear t. cbn in t'. unfold horcomp_data in t'. rewrite id_left in t'. exact t'. 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.