Proposition monad_type_prod_laws : monad_algebra_laws monad_type_prod_map. Proof. split. - use (ProductArrow_eq _ _ _ prod). intro i. rewrite id_left. rewrite !assoc'. etrans. { apply maponpaths. apply ProductPrCommutes. } rewrite !assoc. rewrite <- (nat_trans_ax (η M)). rewrite !assoc'. rewrite (monad_algebra_unit (p i)). rewrite id_right. apply idpath. - use (ProductArrow_eq _ _ _ prod). intro i. rewrite !assoc'. etrans. { apply maponpaths. apply ProductPrCommutes. } refine (!_). etrans. { apply maponpaths. apply ProductPrCommutes. } rewrite !assoc. rewrite <- functor_comp. etrans. { apply maponpaths_2. apply maponpaths. apply ProductPrCommutes. } rewrite <- (nat_trans_ax (μ M)). rewrite functor_comp. rewrite !assoc'. rewrite (monad_algebra_mu (p i)). apply idpath. Qed.
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.Adjunctions.Core. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits. Require Import UniMath.CategoryTheory.Monads.Monads.