Proposition internal_pre_post_comp_as_post_pre_comp {x₁ x₂ : V} (f : x₁ --> x₂) {y₁ y₂ : V} (g : y₁ --> y₂) : internal_pre_post_comp f g = internal_postcomp x₂ g · internal_precomp f y₂. Proof. use internal_funext. intros a h. rewrite tensor_split. rewrite tensor_comp_r_id_r. rewrite !assoc'. unfold internal_precomp, internal_postcomp, internal_pre_post_comp. rewrite !internal_beta. refine (!_). rewrite tensor_split. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_split'. rewrite tensor_split. rewrite !assoc'. rewrite internal_beta. 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.