proposition: Lemma unit_relativelaxcommutator_tensor: relativelaxcommutator_tensor unit_relativelaxcommutator_data.
proof: 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.
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