Definition natural_hom_weq {C D : precategory} (F : functor C D) (G : functor D C) : UU := ∑ (hom_weq : ∏ {A : C} {B : D}, F A --> B ≃ A --> G B), (∏ (A : C) (B : D) (f : F A --> B) (X : C) (h : X --> A), hom_weq (#F h · f) = h · hom_weq f) × (∏ (A : C) (B : D) (f : F A --> B) (Y : D) (k : B --> Y), hom_weq (f · k) = hom_weq f · #G k).
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.