Lemma unit_relativelaxcommutator_unit: relativelaxcommutator_unit unit_relativelaxcommutator_data. Proof. unfold relativelaxcommutator_unit, unit_relativelaxcommutator_data. etrans. 2: { do 2 apply cancel_postcomposition. rewrite unitors_coincide_on_unit. apply pathsinv0, monoidal_rightunitornat. } repeat rewrite assoc'. apply maponpaths. etrans. 2: { apply maponpaths. rewrite <- unitorsinv_coincide_on_unit. apply pathsinv0, monoidal_leftunitorinvnat. } rewrite assoc. etrans. 2: {apply cancel_postcomposition. apply pathsinv0, (pr22 (fmonoidal_preservesunitstrongly U)). } apply pathsinv0, id_left. 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.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Actegories.Actegories. Require Import UniMath.CategoryTheory.Actegories.MorphismsOfActegories. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories. Require Import UniMath.CategoryTheory.coslicecat. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Actegories.CoproductsInActegories. Require Import UniMath.CategoryTheory.Actegories.ProductsInActegories. Require Import UniMath.CategoryTheory.Actegories.ProductActegory. Import BifunctorNotations. Import MonoidalNotations. Import ActegoryNotations. Section ActegoryMorphismFromRelativeLaxCommutator. End ActegoryMorphismFromRelativeLaxCommutator.