Lemma nathomweq_after_adj (H : natural_hom_weq F G) : nathomweq_from_adj (adj_from_nathomweq H) = H. Proof. apply subtypePath. - intros p. apply isapropdirprod. + do 5 (apply impred_isaprop; intro). apply C. + do 5 (apply impred_isaprop; intro). apply C. - cbn. unfold adjunction_hom_weq. do 2 (apply funextsec; intro). apply subtypePath. + intro. apply isapropisweq. + cbn. unfold φ_adj, adj_from_nathomweq. cbn. apply funextsec. intro f. rewrite <- hom_natural_postcomp. apply maponpaths. apply id_left. Defined.
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.