Lemma is_split_monic_to_precomp_is_surjection (m : A --> B) : is_split_monic m -> ∏ c : ob C, issurjective (@precomp_with _ _ _ m c). Proof. intros is_split c f. apply hinhpr. unfold hfiber, precomp_with. exists (pr1 is_split · f). refine (assoc _ _ _ @ _). refine (maponpaths (fun z => z · _) (pr2 is_split) @ _). apply id_left. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Retracts.