Definition full_image_inclusion_commute {C₁ C₂ : category} (F : C₁ ⟶ C₂) : functor_full_img F ∙ sub_precategory_inclusion C₂ (full_img_sub_precategory F) ⟹ F. Proof. use make_nat_trans. - exact (λ _, identity _). - abstract (intros x y f ; cbn ; rewrite id_left, id_right ; apply idpath). Defined.
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.