Lemma φ_adj_inv_from_partial (f : X⟦x, G a⟧) : φ_adj_inv Adj f = pr11 (Huniv x a f). Proof. refine (base_paths _ _ (pr2 (Huniv x a 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_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.