Lemma preIt_ok : # L inF· preIt = ψ μF preIt. Proof. apply cancel_φ. rewrite φ_ψ_μF_eq. rewrite (φ_adj_natural_precomp (pr2 is_left_adj_L)). unfold preIt. rewrite (φ_adj_after_φ_adj_inv (pr2 is_left_adj_L)). rewrite (φ_adj_after_φ_adj_inv (pr2 is_left_adj_L)). assert (iter_eq := algebra_mor_commutes _ _ _ (InitialArrow μF_Initial ⟨_,φ (ψ (R X) (ε X))⟩)). exact iter_eq. 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.