Definition preserves_tensor_nat {C D : category} {TC : tensor C} {TD : tensor D} {F : functor C D} (ptF : preserves_tensor_data TC TD F) : UU := ∏ (x1 x2 y1 y2 : C) (f : C⟦x1,x2⟧) (g : C⟦y1,y2⟧), (ptF x1 y1) · (functor_on_morphisms F (f ⊗^{TC} g)) = ((functor_on_morphisms F f) ⊗^{TD} (functor_on_morphisms F g)) · ptF x2 y2.
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. Import TensorNotation. Import TensorUnitNotation. Import TensorNotation. Import TensorUnitNotation. Import TensorNotation. Import TensorUnitNotation. Import TensorNotation. Import TensorUnitNotation. Import UnitorsAssociatorNotation. Import TensorNotation. Import TensorUnitNotation. Import UnitorsAssociatorNotation.