Lemma lineatorinv_nat_left {ld : lineator_data} (ls : lineator_strongly ld) (lln : lineator_nat_left ld) (v : V) (x1 x2 : C) (f : C⟦x1,x2⟧) : (is_z_isomorphism_mor (ls v x1)) · v ⊗^{ActD}_{l} # F f = # F (v ⊗^{ActC}_{l} f) · (is_z_isomorphism_mor (ls v x2)). Proof. set (ldiso := ld v x1 ,, ls v x1 : z_iso _ _). apply (z_iso_inv_on_right _ _ _ ldiso). rewrite assoc. etrans. 2: { apply maponpaths_2. apply lln. } rewrite assoc'. unfold is_z_isomorphism_mor. rewrite (pr12 (ls v x2)). apply (! id_right _). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.CategoryTheory.Actegories.Actegories. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories. Require Import UniMath.CategoryTheory.coslicecat. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects. Import BifunctorNotations. Import MonoidalNotations. Import ActegoryNotations.