proposition: Lemma fmonoidal_preservesassociativityinv {C D : category} {M : monoidal C} {N : monoidal D} {F : functor C D} (fm : fmonoidal_lax M N F) : preserves_associativityinv (fmonoidal_preservestensordata fm).
proof: Proof. intros x y z. rewrite assoc'. apply (z_iso_inv_on_right _ _ _ (_,,_,, monoidal_associatorisolaw N _ _ _)). cbn. etrans. 2: { repeat rewrite assoc. apply cancel_postcomposition. apply fmonoidal_preservesassociativity. } repeat rewrite assoc'. apply maponpaths. etrans. 2: { apply maponpaths. rewrite <- functor_comp. apply maponpaths. apply pathsinv0, (monoidal_associatorisolaw M). } rewrite functor_id. 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.whiskering UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Structure.Symmetric