Definition regular_epi_in_slice_weq {C : category} (PB : Pullbacks C) (EC : coeqs_of_kernel_pair C) (HC : regular_epi_pb_stable C) {x : C} {yg₁ zg₂ : C/x} (hp : yg₁ --> zg₂) : is_regular_epi hp ≃ is_regular_epi (pr1 hp). Proof. use weqimplimpl. - apply (from_regular_epi_in_slice PB EC HC). - apply to_regular_epi_in_slice. - apply isaprop_is_regular_epi. - apply isaprop_is_regular_epi. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.EpiFacts. Require Import UniMath.CategoryTheory.RegularAndExact.RegularEpi. Require Import UniMath.CategoryTheory.RegularAndExact.RegularCategory. Require Import UniMath.CategoryTheory.RegularAndExact.RegularEpiFacts. Require Import UniMath.CategoryTheory.RegularAndExact.ExactCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodColimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodMorphisms. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor. Section PreservesFromStable. End PreservesFromStable.