Lemma unit_relativelaxcommutator_tensor: relativelaxcommutator_tensor unit_relativelaxcommutator_data. Proof. intro; intros. unfold relativelaxcommutator_tensor_body, unit_relativelaxcommutator_data. etrans. 2: { do 2 apply cancel_postcomposition. etrans. 2: { do 2 apply cancel_postcomposition. rewrite assoc'. do 2 apply maponpaths. apply pathsinv0, (functor_comp (leftwhiskering_functor Mon_V _)). } apply maponpaths. apply pathsinv0, (functor_comp (rightwhiskering_functor Mon_V _)). } cbn. etrans. 2: { repeat rewrite assoc'. apply maponpaths. repeat rewrite assoc. do 6 apply cancel_postcomposition. apply pathsinv0, left_whisker_with_runitor. } etrans. 2: { repeat rewrite assoc. do 4 apply cancel_postcomposition. repeat rewrite assoc'. do 2 apply maponpaths. apply pathsinv0, monoidal_triangle_identity_inv. } etrans. 2: { repeat rewrite assoc'. do 2 apply maponpaths. repeat rewrite assoc. do 3 apply cancel_postcomposition. etrans. 2: { apply (functor_comp (rightwhiskering_functor Mon_V _)). } apply maponpaths. apply pathsinv0, monoidal_rightunitorisolaw. } rewrite functor_id, id_left. etrans. 2: { do 2 apply maponpaths. apply cancel_postcomposition. rewrite <- monoidal_triangle_identity'_inv. rewrite assoc'. apply maponpaths. apply pathsinv0, monoidal_associatorisolaw. } rewrite id_right. etrans. 2: { repeat rewrite assoc'. do 2 apply maponpaths. apply pathsinv0, monoidal_leftunitorinvnat. } do 2 rewrite assoc. apply cancel_postcomposition. etrans. 2: { apply cancel_postcomposition. apply pathsinv0, monoidal_rightunitornat. } etrans. 2: { rewrite assoc'. apply maponpaths. apply pathsinv0, fmonoidal_preservestensorstrongly. } apply pathsinv0, 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.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.