Lemma cartesian_rinvunitor' (x : V) : diag x · x ⊗^{ M}_{l} aug x = mon_rinvunitor x. Proof. etrans. { refine (_ @ postcompWithBinProductArrow _ (is_cartesian_BinProduct Ccart _ _) (is_cartesian_BinProduct Ccart _ _) (identity x) (semi_cart_to_unit Ccart x) (identity x) (identity x)). apply maponpaths. rewrite <- (when_bifunctor_becomes_leftwhiskering M). exact (cartesian_tensor_mor Ccart (identity x) (semi_cart_to_unit Ccart x)). } do 2 rewrite id_left. exact (cartesian_rinvunitor x). 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.catiso. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Projection. Require Import UniMath.CategoryTheory.DisplayedCats.CatIsoDisplayed. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import BifunctorNotations. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Category. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Tensor. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Symmetric. Import MonoidalNotations. Import ComonoidNotations. End CartesianFromComonoidsToCartesian. *)