Definition regular_epi_mono_factorization : ∑ (im : C) (e : x --> im) (m : im --> y), is_regular_epi e × isMonic m × f = e · m. Proof. simple refine (_ ,, e ,, m ,, _ ,, _ ,, _). - exact regular_epi_mono_factorization_is_regular_epi. - exact regular_epi_mono_factorization_is_regular_is_monic. - exact regular_epi_mono_factorization_comm. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.EpiFacts. Require Import UniMath.CategoryTheory.RegularAndExact.RegularEpi.