Lemma id_is_monoid_mor {x : C} (xx : monoid x) : is_monoid_mor xx xx (identity x). Proof. split. - refine (_ @ ! id_right _). etrans. { apply maponpaths_2, bifunctor_distributes_over_id. apply (bifunctor_leftid M). apply (bifunctor_rightid M). } apply id_left. - apply id_right. 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.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import BifunctorNotations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total.