Definition monad_to_hset_struct_adj_lift : monad_free_alg X --> YY. Proof. refine (#M f · hY ,, _). cbn. use funextsec. intro x. etrans. { apply maponpaths. exact (!(eqtohomot (nat_trans_ax (μ M) _ _ f) x)). } cbn. refine (eqtohomot (monad_algebra_mu hY) (# M (# M f) x) @ _). cbn. apply maponpaths. exact (!(eqtohomot (functor_comp M _ _) _)). Defined.
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.