Lemma pre_composition_with_ess_surj_and_fully_faithful_is_fully_faithful (p : essentially_surjective H) (Hff : fully_faithful H): fully_faithful (pre_composition_functor A B C H). Proof. apply pre_composition_with_ess_surj_and_full_is_fully_faithful. - exact p. - apply fully_faithful_implies_full_and_faithful. exact Hff. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering.