Definition functor_tensor_composition {C D E : category} {TC : functor (C ⊠ C) C} {TD : functor (D ⊠ D) D} {TE : functor (E ⊠ E) E} {F : functor C D} {G : functor D E} (FF : functor_tensor TC TD F) (GG : functor_tensor TD TE G) : functor_tensor TC TE (functor_composite F G). Proof. exists (λ cc, GG (F (pr1 cc) , F (pr2 cc)) · #G (FF cc)). apply functor_tensor_composition_is_nat_trans. Defined.
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.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored.