Definition commutative_comonoid_to_comonoid_of_comonoids_data (m : commutative_comonoid V) : comonoid_data (symmetric_cat_commutative_comonoids V). Proof. exists m. refine (_ ,, _). - refine (δ_{m} ,, (_ ,, tt) ,, tt). abstract (split ; cbn; [ refine (! commutative_symmetric_braiding_after_4_copies V m @ _); apply maponpaths; cbn; unfold dialgebra_disp_tensor_op; apply maponpaths_2; apply pathsinv0, id_left | refine (id_right _ @ ! diagonal_is_comonoid_mor_counit m @ _); apply maponpaths; unfold dialgebra_disp_tensor_op; cbn; apply maponpaths_2; apply pathsinv0, id_left]). - refine (ε_{m} ,, (_ ,, tt) ,, tt). abstract (split ; cbn; [ refine (aug_is_comonoid_mor_comult m @ _); apply maponpaths; apply pathsinv0, id_left | refine (_ @ assoc' _ _ _); apply pathsinv0, id_right]). Defined.
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.Limits.BinProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Core. 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.Displayed.Symmetric. Require Import UniMath.CategoryTheory.Categories.Dialgebras. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalDialgebras. 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. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.MonoidalCartesianBuilder. Import MonoidalNotations. Import ComonoidNotations.