Lemma functor_on_inv_from_iso' {C C' : precategory} (F : functor C C') {a b : ob C} {f : a --> b} (H : is_iso f) : inv_from_iso (make_iso _ (functor_on_is_iso_is_iso F H)) = # F (inv_from_iso (make_iso _ H)). Proof. apply pathsinv0. use inv_iso_unique'. cbn. unfold precomp_with. rewrite <- functor_comp. set (tmp := iso_inv_after_iso (make_iso _ H)). cbn in tmp. rewrite tmp. apply functor_id. Qed.
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.