Lemma φ_ψ_μF_eq (h: L μF --> X): φ (ψ μF h) = #F (φ h) · φ(ψ (R X) (ε X)). Proof. rewrite <- (φ_adj_natural_precomp (pr2 is_left_adj_L)). apply maponpaths. eapply pathscomp0. 2: apply ψ_naturality. apply maponpaths. rewrite truth_about_ε. rewrite <- (φ_adj_inv_natural_precomp (pr2 is_left_adj_L)). rewrite id_right. apply pathsinv0. change (φ_inv(φ h) = h). apply φ_adj_inv_after_φ_adj. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.Adjunctions.Core.