Lemma is_nat_trans_tensor_comp {F G H : functor C D} (FF : functor_tensor F) (GG : functor_tensor G) (HH : functor_tensor H) {α : nat_trans F G} {β : nat_trans G H} (αα : is_nat_trans_tensor FF GG α) (ββ : is_nat_trans_tensor GG HH β) : is_nat_trans_tensor FF HH (nat_trans_comp _ _ _ α β). Proof. intros x y. simpl. etrans. { apply maponpaths_2. exact (functor_comp TD (α x #, α y) (β x #, β y)). } rewrite assoc'. etrans. { apply maponpaths. exact (ββ x y). } do 2 rewrite assoc. apply maponpaths_2. apply αα. 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.