Lemma φ_adj_natural_precomp (A : C) (B : D) (f : F A --> B) (X : C) (h : X --> A) : φ_adj (#F h · f) = h · φ_adj f. Proof. unfold φ_adj. rewrite functor_comp. set (T:=nat_trans_ax η); simpl in T. rewrite assoc. rewrite <- T. apply pathsinv0, assoc. 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.