Lemma cobind_cobind {a b c : C} (f : C⟦T a,b⟧) (g : C⟦T b,c⟧) : cobind f · cobind g = cobind (cobind f · g). Proof. unfold cobind; rewrite assoc. rewrite !functor_comp. repeat rewrite assoc. apply cancel_postcomposition. etrans. { rewrite assoc'; apply maponpaths, (nat_trans_ax (δ T) _ _ f). } rewrite assoc. apply cancel_postcomposition. apply (!Comonad_law3 a). Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.SIP. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.Core.Univalence.