Lemma is_inverse_functor_image (C C' : precategory) (F : functor C C') (a b : C) (f : iso a b): is_inverse_in_precat (#F f) (#F (inv_from_iso f)). Proof. simpl; split; simpl. rewrite <- functor_comp. rewrite iso_inv_after_iso. apply functor_id. rewrite <- functor_comp. rewrite iso_after_iso_inv. 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.