Lemma preservestensor_is_nattrans_full {C D : category} {M : monoidal C} {N : monoidal D} {F : functor C D} {pt : preserves_tensordata M N F} (ptnl : preserves_tensor_nat_left pt) (ptnr : preserves_tensor_nat_right pt) : ∏ (x1 x2 y1 y2 : C) (f : C⟦x1,x2⟧) (g : C⟦y1,y2⟧), # F f ⊗^{ N} # F g · pt x2 y2 = pt x1 y1 · # F (f ⊗^{ M} g). Proof. intros. etrans. { unfold functoronmorphisms1. rewrite assoc'. rewrite ptnl. apply assoc. } rewrite ptnr. rewrite assoc'. apply maponpaths. apply pathsinv0, functor_comp. 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.