Lemma functor_on_iso_inv (C C' : precategory) (F : functor C C') (a b : ob C) (f : iso a b) : functor_on_iso F (iso_inv_from_iso f) = iso_inv_from_iso (functor_on_iso F f). Proof. apply eq_iso; simpl. apply inv_iso_unique'; simpl. unfold precomp_with. rewrite <- functor_comp. rewrite iso_inv_after_iso. apply functor_id. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Core.Univalence.