Lemma fmonoidal_preservesrightunitalityinv {C D : category} {M : monoidal C} {N : monoidal D} {F : functor C D} (fm : fmonoidal_lax M N F) : preserves_rightunitalityinv (fmonoidal_preservestensordata fm) (fmonoidal_preservesunit fm). Proof. intro x. rewrite assoc'. apply (z_iso_inv_on_right _ _ _ (_,,_,, monoidal_rightunitorisolaw N (F x))). cbn. rewrite <- (fmonoidal_preservesrightunitality fm). repeat rewrite assoc'. apply maponpaths. etrans. 2: { apply maponpaths. apply functor_comp. } etrans. 2: { do 2 apply maponpaths. apply pathsinv0, (pr1(monoidal_rightunitorisolaw M x)). } rewrite functor_id. 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.whiskering. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import BifunctorNotations. Import MonoidalNotations.