Lemma nat_trans_tensor_ntrans1_is_nat_trans : is_nat_trans (functor_tensor_map_dom TD F) (functor_tensor_map_codom TC G) (λ cc : C × C, # TD (α (pr1 cc) #, α (pr2 cc)) · GG cc). Proof. intros cc1 cc2 ff. simpl. rewrite assoc. rewrite <- (functor_comp TD). rewrite <- binprod_comp. do 2 rewrite (pr2 α). rewrite binprod_comp. rewrite (functor_comp TD). do 2 rewrite assoc'. apply maponpaths. apply (pr2 GG). 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.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.