proposition: Lemma composedrelativelaxcommutator_nat: relativelaxcommutator_nat composedrelativelaxcommutator_data.
proof: Proof. do 2 red; intros; unfold composedrelativelaxcommutator_data; cbn. assert (γ1_nat := relativelaxcommutator_ldnat γ1). assert (γ2_nat := relativelaxcommutator_ldnat γ2). do 2 red in γ1_nat, γ2_nat; cbn in γ1_nat, γ2_nat. etrans. { repeat rewrite assoc. do 4 apply cancel_postcomposition. apply monoidal_associatorinvnatright. } repeat rewrite assoc'. apply maponpaths. etrans. { rewrite assoc. apply cancel_postcomposition. apply pathsinv0, (functor_comp (rightwhiskering_functor Mon_V v2)). } cbn. rewrite γ1_nat. etrans. { rewrite assoc. do 2 apply cancel_postcomposition. apply (functor_comp (rightwhiskering_functor Mon_V v2)). } cbn. repeat rewrite assoc'. apply maponpaths. etrans. 2: { do 2 apply maponpaths. apply monoidal_associatorinvnatleft. } repeat rewrite assoc. apply cancel_postcomposition. etrans. 2: { rewrite assoc'. apply maponpaths. apply (functor_comp (leftwhiskering_functor Mon_V v1)). } cbn. rewrite <- γ2_nat. etrans. 2: { apply maponpaths. apply pathsinv0, (functor_comp (leftwhiskering_functor Mon_V v1)). } cbn. repeat rewrite assoc. apply cancel_postcomposition. apply pathsinv0, monoidal_associatornatleftright. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Actegories.Actegories UniMath.CategoryTheory.Actegories.MorphismsOfActegories UniMath.CategoryTheory.Actegories.ConstructionOfActegories UniMath.CategoryTheory.coslicecat UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Actegories.CoproductsInActegories UniMath.CategoryTheory.Actegories.ProductsInActegories UniMath.CategoryTheory.Actegories.ProductActegory