Lemma truth_about_ε (A: C'): ε A = φ_inv (identity (R A)). Proof. unfold φ_inv, φ_adj_inv. rewrite functor_id. apply pathsinv0. apply id_left. 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.