Lemma is_functor_full_img (C D: category) (F : functor C D) : is_functor (full_img_functor_data F). Proof. split. intro a; simpl. apply subtypePath. intro; apply pr2. apply functor_id. intros a b c f g. set ( H := eq_in_sub_precategory D (full_img_sub_precategory F)). apply (H (full_img_functor_obj F a)(full_img_functor_obj F c)). simpl; apply functor_comp. Qed.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Subcategory.Core.