proposition: Lemma fmonoidal_preservesleftunitalityinv {C D : category} {M : monoidal C} {N : monoidal D} {F : functor C D} (fm : fmonoidal_lax M N F) : preserves_leftunitalityinv (fmonoidal_preservestensordata fm) (fmonoidal_preservesunit fm).
proof: Proof. intro x. rewrite assoc'. apply (z_iso_inv_on_right _ _ _ (_,,_,, monoidal_leftunitorisolaw N (F x))). cbn. rewrite <- (fmonoidal_preservesleftunitality fm). repeat rewrite assoc'. apply maponpaths. etrans. 2: { apply maponpaths. apply functor_comp. } etrans. 2: { do 2 apply maponpaths. apply pathsinv0, (pr1(monoidal_leftunitorisolaw M x)). } 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