Lemma preserves_tensorinv_nat_right {C D : category} {M : monoidal C} {N : monoidal D} {F : functor C D} {pt : preserves_tensordata M N F} (pts : preserves_tensor_strongly pt) (ptrn : preserves_tensor_nat_right pt) (x1 x2 y : C) (f : C⟦x1,x2⟧) : (is_z_isomorphism_mor (pts x1 y)) · # F f ⊗^{ N}_{r} F y = # F (f ⊗^{ M}_{r} y) · (is_z_isomorphism_mor (pts x2 y)). Proof. set (ptiso := pt x1 y ,, pts x1 y : z_iso _ _). apply (z_iso_inv_on_right _ _ _ ptiso). rewrite assoc. etrans. 2: { apply maponpaths_2. apply ptrn. } rewrite assoc'. unfold is_z_isomorphism_mor. rewrite (pr12 (pts x2 y)). apply (! 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.