Lemma ff_essentially_surjective_to_split (A B : category) (HA : is_univalent A) (F : functor A B) (HF : fully_faithful F) (HF' : essentially_surjective F) : split_essentially_surjective F. Proof. intro b. apply (squash_to_prop (HF' b)). - apply isaprop_sigma_z_iso; assumption. - exact (idfun _). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Adjunctions.Core.