Lemma φ_adj_from_partial (f : A⟦F x, a⟧) : φ_adj Adj f = pr11 (Huniv a x f). Proof. refine (base_paths _ _ (pr2 (Huniv a x f) (_ ,, !_))). refine (maponpaths (λ x, x · _) (functor_comp _ _ _) @ _). refine (assoc' _ _ _ @ _). refine (!maponpaths (λ x, _ · x) (pr21 (Huniv _ _ _)) @ _). refine (assoc _ _ _ @ _). refine (!maponpaths (λ x, x · _) (pr21 (Huniv _ _ _)) @ _). apply id_left. 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.