Lemma φ_adj_after_φ_adj_inv {A : C} {B : D} (g : A --> G B) : φ_adj (φ_adj_inv g) = g. Proof. unfold φ_adj. unfold φ_adj_inv. assert (X':=triangle_id_right_ad H). rewrite functor_comp. rewrite assoc. assert (X2 := nat_trans_ax η). simpl in X2. rewrite <- X2; clear X2. rewrite <- assoc. intermediate_path (g · identity _). - apply maponpaths. apply X'. - apply id_right. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.SplitMonicsAndEpis. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Section HomSetIsoRightFromPartial. End HomSetIsoRightFromPartial. Section HomSetIsoLeftFromPartial. End HomSetIsoLeftFromPartial.