Lemma is_linear_nat_trans_pointwise_inverse {F G : functor C D} (Fl : lineator_lax Mon_V ActC ActD F) (Gl : lineator_lax Mon_V ActC ActD G) (ξ : F ⟹ G) (isnziξ : is_nat_z_iso ξ) : is_linear_nat_trans Fl Gl ξ -> is_linear_nat_trans Gl Fl (nat_z_iso_inv (ξ,,isnziξ)). Proof. intros islnt v x. cbn. set (aux := (_,,is_z_iso_leftwhiskering_z_iso (actegory_action Mon_V ActD) v (ξ x) (isnziξ x) : z_iso _ _)). apply pathsinv0, (z_iso_inv_on_right _ _ _ aux). rewrite assoc. apply (z_iso_inv_on_left _ _ _ _ (_,,isnziξ (v ⊗_{ActC} x))). cbn. apply (!(islnt v x)). 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.