Definition initial_functor_endo_is_identity {A : category} (F : initial_functor_from A) (G : target_category F ⟶ target_category F) (H : F ∙ G = F) : G = functor_identity (target_category F). Proof. apply (uniqueExists (initial_functor_from_is_initial F F)). - exact H. - apply functor_identity_right. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.PrecompEquivalence. Require Import UniMath.CategoryTheory.WeakEquivalences.Core. Require Import UniMath.CategoryTheory.whiskering.