Lemma lineator_preservesunitorinv (ll : lineator_lax) : preserves_unitorinv ll. Proof. intro x. apply (z_iso_inv_on_right _ _ _ (_,,_,, actegory_unitorisolaw Mon_V ActD (F x))). cbn. rewrite <- (lineator_preservesunitor ll). repeat rewrite assoc'. etrans. { apply pathsinv0, id_right. } apply maponpaths. etrans. 2: { rewrite <- functor_comp. apply maponpaths. apply pathsinv0, (actegory_unitorisolaw Mon_V ActC x). } apply pathsinv0, functor_id. 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.