Proposition internal_precomp_comp {x₁ x₂ x₃ : V} (f₁ : x₁ --> x₂) (f₂ : x₂ --> x₃) (y : V) : internal_precomp (f₁ · f₂) y = internal_precomp f₂ y · internal_precomp f₁ y. Proof. use internal_funext. intros a h. rewrite tensor_split. rewrite tensor_comp_r_id_r. rewrite !assoc'. unfold internal_precomp. rewrite !internal_beta. refine (!_). rewrite tensor_split. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_split'. rewrite tensor_split. rewrite !assoc'. rewrite internal_beta. rewrite !assoc. rewrite <- tensor_comp_id_l. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import MonoidalNotations.