proposition: Lemma is_functor_full_img (C D: category) (F : functor C D) : is_functor (full_img_functor_data F).
proof: 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.
imports: UniMath.Foundations.Sets UniMath.MoreFoundations.PartA UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Subcategory.Core