/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Jakob von Raumer

! This file was ported from Lean 3 source module category_theory.preadditive.basic
! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Algebra.BigOperators.Basic
import Mathbin.Algebra.Hom.Group
import Mathbin.Algebra.Module.Basic
import Mathbin.CategoryTheory.Endomorphism
import Mathbin.CategoryTheory.Limits.Shapes.Kernels

/-!
# Preadditive categories

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

A preadditive category is a category in which `X ⟶ Y` is an abelian group in such a way that
composition of morphisms is linear in both variables.

This file contains a definition of preadditive category that directly encodes the definition given
above. The definition could also be phrased as follows: A preadditive category is a category
enriched over the category of Abelian groups. Once the general framework to state this in Lean is
available, the contents of this file should become obsolete.

## Main results

* Definition of preadditive categories and basic properties
* In a preadditive category, `f : Q ⟶ R` is mono if and only if `g ≫ f = 0 → g = 0` for all
  composable `g`.
* A preadditive category with kernels has equalizers.

## Implementation notes

The simp normal form for negation and composition is to push negations as far as possible to
the outside. For example, `f ≫ (-g)` and `(-f) ≫ g` both become `-(f ≫ g)`, and `(-f) ≫ (-g)`
is simplified to `f ≫ g`.

## References

* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]

## Tags

additive, preadditive, Hom group, Ab-category, Ab-enriched
-/

universe v u

open CategoryTheory.Limits

open BigOperators

namespace CategoryTheory

variable (C : Type u) [Category.{v} C]

#print CategoryTheory.Preadditive /-
/-- A category is called preadditive if `P ⟶ Q` is an abelian group such that composition is
    linear in both variables. -/
class Preadditive where
  homGroup : ∀ P Q : C, AddCommGroup (P ⟶ Q) := by infer_instance
  add_comp : ∀ (P Q R : C) (f f' : P ⟶ Q) (g : Q ⟶ R), (f + f') ≫ g = f ≫ g + f' ≫ g := by obviously
  comp_add : ∀ (P Q R : C) (f : P ⟶ Q) (g g' : Q ⟶ R), f ≫ (g + g') = f ≫ g + f ≫ g' := by obviously
#align category_theory.preadditive CategoryTheory.Preadditive
-/

attribute [instance] preadditive.hom_group

restate_axiom preadditive.add_comp'

restate_axiom preadditive.comp_add'

attribute [simp, reassoc.1] preadditive.add_comp

attribute [reassoc.1] preadditive.comp_add

-- (the linter doesn't like `simp` on this lemma)
attribute [simp] preadditive.comp_add

end CategoryTheory

open CategoryTheory

namespace CategoryTheory

namespace Preadditive

section Preadditive

open AddMonoidHom

variable {C : Type u} [Category.{v} C] [Preadditive C]

section InducedCategory

universe u'

variable {C} {D : Type u'} (F : D → C)

#print CategoryTheory.Preadditive.inducedCategory /-
instance inducedCategory : Preadditive.{v} (InducedCategory C F)
    where
  homGroup P Q := @Preadditive.homGroup C _ _ (F P) (F Q)
  add_comp P Q R f f' g := add_comp _ _ _ _ _ _
  comp_add P Q R f g g' := comp_add _ _ _ _ _ _
#align category_theory.preadditive.induced_category CategoryTheory.Preadditive.inducedCategory
-/

end InducedCategory

/- warning: category_theory.preadditive.full_subcategory -> CategoryTheory.Preadditive.fullSubcategory is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] (Z : C -> Prop), CategoryTheory.Preadditive.{u1, u2} (CategoryTheory.FullSubcategoryₓ.{u1, u2} C _inst_1 Z) (CategoryTheory.FullSubcategory.category.{u1, u2} C _inst_1 Z)
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] (Z : C -> Prop), CategoryTheory.Preadditive.{u1, u2} (CategoryTheory.FullSubcategory.{u2} C Z) (CategoryTheory.FullSubcategory.category.{u1, u2} C _inst_1 Z)
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.full_subcategory CategoryTheory.Preadditive.fullSubcategoryₓ'. -/
instance fullSubcategory (Z : C → Prop) : Preadditive.{v} (FullSubcategory Z)
    where
  homGroup P Q := @Preadditive.homGroup C _ _ P.obj Q.obj
  add_comp P Q R f f' g := add_comp _ _ _ _ _ _
  comp_add P Q R f g g' := comp_add _ _ _ _ _ _
#align category_theory.preadditive.full_subcategory CategoryTheory.Preadditive.fullSubcategory

instance (X : C) : AddCommGroup (End X) := by
  dsimp [End]
  infer_instance

instance (X : C) : Ring (End X) :=
  { (inferInstance : AddCommGroup (End X)),
    (inferInstance :
      Monoid (End
          X)) with
    left_distrib := fun f g h => Preadditive.add_comp X X X g h f
    right_distrib := fun f g h => Preadditive.comp_add X X X h f g }

#print CategoryTheory.Preadditive.leftComp /-
/-- Composition by a fixed left argument as a group homomorphism -/
def leftComp {P Q : C} (R : C) (f : P ⟶ Q) : (Q ⟶ R) →+ (P ⟶ R) :=
  mk' (fun g => f ≫ g) fun g g' => by simp
#align category_theory.preadditive.left_comp CategoryTheory.Preadditive.leftComp
-/

#print CategoryTheory.Preadditive.rightComp /-
/-- Composition by a fixed right argument as a group homomorphism -/
def rightComp (P : C) {Q R : C} (g : Q ⟶ R) : (P ⟶ Q) →+ (P ⟶ R) :=
  mk' (fun f => f ≫ g) fun f f' => by simp
#align category_theory.preadditive.right_comp CategoryTheory.Preadditive.rightComp
-/

variable {P Q R : C} (f f' : P ⟶ Q) (g g' : Q ⟶ R)

#print CategoryTheory.Preadditive.compHom /-
/-- Composition as a bilinear group homomorphism -/
def compHom : (P ⟶ Q) →+ (Q ⟶ R) →+ (P ⟶ R) :=
  AddMonoidHom.mk' (fun f => leftComp _ f) fun f₁ f₂ =>
    AddMonoidHom.ext fun g => (rightComp _ g).map_add f₁ f₂
#align category_theory.preadditive.comp_hom CategoryTheory.Preadditive.compHom
-/

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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.sub_comp CategoryTheory.Preadditive.sub_compₓ'. -/
@[simp, reassoc.1]
theorem sub_comp : (f - f') ≫ g = f ≫ g - f' ≫ g :=
  map_sub (rightComp P g) f f'
#align category_theory.preadditive.sub_comp CategoryTheory.Preadditive.sub_comp

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but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {P : C} {Q : C} {R : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (g' : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R f (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 Q R))))) g g')) (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 P R))))) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R f g) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R f g'))
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.comp_sub CategoryTheory.Preadditive.comp_subₓ'. -/
-- The redundant simp lemma linter says that simp can prove the reassoc version of this lemma.
@[reassoc.1, simp]
theorem comp_sub : f ≫ (g - g') = f ≫ g - f ≫ g' :=
  map_sub (leftComp R f) g g'
#align category_theory.preadditive.comp_sub CategoryTheory.Preadditive.comp_sub

/- warning: category_theory.preadditive.neg_comp -> CategoryTheory.Preadditive.neg_comp is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.neg_comp CategoryTheory.Preadditive.neg_compₓ'. -/
@[simp, reassoc.1]
theorem neg_comp : (-f) ≫ g = -f ≫ g :=
  map_neg (rightComp P g) f
#align category_theory.preadditive.neg_comp CategoryTheory.Preadditive.neg_comp

/- warning: category_theory.preadditive.comp_neg -> CategoryTheory.Preadditive.comp_neg is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {P : C} {Q : C} {R : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R f (Neg.neg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (SubNegMonoid.toHasNeg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 Q R)))) g)) (Neg.neg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (SubNegMonoid.toHasNeg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 P R)))) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R f g))
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.comp_neg CategoryTheory.Preadditive.comp_negₓ'. -/
-- The redundant simp lemma linter says that simp can prove the reassoc version of this lemma.
@[reassoc.1, simp]
theorem comp_neg : f ≫ (-g) = -f ≫ g :=
  map_neg (leftComp R f) g
#align category_theory.preadditive.comp_neg CategoryTheory.Preadditive.comp_neg

/- warning: category_theory.preadditive.neg_comp_neg -> CategoryTheory.Preadditive.neg_comp_neg is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {P : C} {Q : C} {R : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R (Neg.neg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (SubNegMonoid.toHasNeg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 P Q)))) f) (Neg.neg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (SubNegMonoid.toHasNeg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 Q R)))) g)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R f g)
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {P : C} {Q : C} {R : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R (Neg.neg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (NegZeroClass.toNeg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (SubNegZeroMonoid.toNegZeroClass.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (SubtractionMonoid.toSubNegZeroMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (SubtractionCommMonoid.toSubtractionMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (AddCommGroup.toDivisionAddCommMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 P Q)))))) f) (Neg.neg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (NegZeroClass.toNeg.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (SubNegZeroMonoid.toNegZeroClass.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (SubtractionMonoid.toSubNegZeroMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (SubtractionCommMonoid.toSubtractionMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (AddCommGroup.toDivisionAddCommMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 Q R)))))) g)) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R f g)
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.neg_comp_neg CategoryTheory.Preadditive.neg_comp_negₓ'. -/
@[reassoc.1]
theorem neg_comp_neg : (-f) ≫ (-g) = f ≫ g := by simp
#align category_theory.preadditive.neg_comp_neg CategoryTheory.Preadditive.neg_comp_neg

#print CategoryTheory.Preadditive.nsmul_comp /-
theorem nsmul_comp (n : ℕ) : (n • f) ≫ g = n • f ≫ g :=
  map_nsmul (rightComp P g) n f
#align category_theory.preadditive.nsmul_comp CategoryTheory.Preadditive.nsmul_comp
-/

#print CategoryTheory.Preadditive.comp_nsmul /-
theorem comp_nsmul (n : ℕ) : f ≫ (n • g) = n • f ≫ g :=
  map_nsmul (leftComp R f) n g
#align category_theory.preadditive.comp_nsmul CategoryTheory.Preadditive.comp_nsmul
-/

#print CategoryTheory.Preadditive.zsmul_comp /-
theorem zsmul_comp (n : ℤ) : (n • f) ≫ g = n • f ≫ g :=
  map_zsmul (rightComp P g) n f
#align category_theory.preadditive.zsmul_comp CategoryTheory.Preadditive.zsmul_comp
-/

#print CategoryTheory.Preadditive.comp_zsmul /-
theorem comp_zsmul (n : ℤ) : f ≫ (n • g) = n • f ≫ g :=
  map_zsmul (leftComp R f) n g
#align category_theory.preadditive.comp_zsmul CategoryTheory.Preadditive.comp_zsmul
-/

/- warning: category_theory.preadditive.comp_sum -> CategoryTheory.Preadditive.comp_sum is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.comp_sum CategoryTheory.Preadditive.comp_sumₓ'. -/
@[reassoc.1]
theorem comp_sum {P Q R : C} {J : Type _} (s : Finset J) (f : P ⟶ Q) (g : J → (Q ⟶ R)) :
    (f ≫ ∑ j in s, g j) = ∑ j in s, f ≫ g j :=
  map_sum (leftComp R f) _ _
#align category_theory.preadditive.comp_sum CategoryTheory.Preadditive.comp_sum

/- warning: category_theory.preadditive.sum_comp -> CategoryTheory.Preadditive.sum_comp is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {P : C} {Q : C} {R : C} {J : Type.{u3}} (s : Finset.{u3} J) (f : J -> (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q)) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Q R), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R (Finset.sum.{u1, u3} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) J (AddCommGroup.toAddCommMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Q) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 P Q)) s (fun (j : J) => f j)) g) (Finset.sum.{u1, u3} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) J (AddCommGroup.toAddCommMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P R) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 P R)) s (fun (j : J) => CategoryTheory.CategoryStruct.comp.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1) P Q R (f j) g))
but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.Preadditive.{u2, u3} C _inst_1] {P : C} {Q : C} {R : C} {J : Type.{u1}} (s : Finset.{u1} J) (f : J -> (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) P Q)) (g : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Q R), Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) P R) (CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) P Q R (Finset.sum.{u2, u1} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) P Q) J (AddCommGroup.toAddCommMonoid.{u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) P Q) (CategoryTheory.Preadditive.homGroup.{u2, u3} C _inst_1 _inst_2 P Q)) s (fun (j : J) => f j)) g) (Finset.sum.{u2, u1} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) P R) J (AddCommGroup.toAddCommMonoid.{u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) P R) (CategoryTheory.Preadditive.homGroup.{u2, u3} C _inst_1 _inst_2 P R)) s (fun (j : J) => CategoryTheory.CategoryStruct.comp.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1) P Q R (f j) g))
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.sum_comp CategoryTheory.Preadditive.sum_compₓ'. -/
@[reassoc.1]
theorem sum_comp {P Q R : C} {J : Type _} (s : Finset J) (f : J → (P ⟶ Q)) (g : Q ⟶ R) :
    (∑ j in s, f j) ≫ g = ∑ j in s, f j ≫ g :=
  map_sum (rightComp P g) _ _
#align category_theory.preadditive.sum_comp CategoryTheory.Preadditive.sum_comp

instance {P Q : C} {f : P ⟶ Q} [Epi f] : Epi (-f) :=
  ⟨fun R g g' H => by rwa [neg_comp, neg_comp, ← comp_neg, ← comp_neg, cancel_epi, neg_inj] at H⟩

instance {P Q : C} {f : P ⟶ Q} [Mono f] : Mono (-f) :=
  ⟨fun R g g' H => by rwa [comp_neg, comp_neg, ← neg_comp, ← neg_comp, cancel_mono, neg_inj] at H⟩

#print CategoryTheory.Preadditive.preadditiveHasZeroMorphisms /-
instance (priority := 100) preadditiveHasZeroMorphisms : HasZeroMorphisms C
    where
  Zero := inferInstance
  comp_zero P Q f R := show leftComp R f 0 = 0 from map_zero _
  zero_comp P Q R f := show rightComp P f 0 = 0 from map_zero _
#align category_theory.preadditive.preadditive_has_zero_morphisms CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
-/

/- warning: category_theory.preadditive.module_End_right -> CategoryTheory.Preadditive.moduleEndRight is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.module_End_right CategoryTheory.Preadditive.moduleEndRightₓ'. -/
instance moduleEndRight {X Y : C} : Module (End Y) (X ⟶ Y)
    where
  smul_add r f g := add_comp _ _ _ _ _ _
  smul_zero r := zero_comp
  add_smul r s f := comp_add _ _ _ _ _ _
  zero_smul r := comp_zero
#align category_theory.preadditive.module_End_right CategoryTheory.Preadditive.moduleEndRight

#print CategoryTheory.Preadditive.mono_of_cancel_zero /-
theorem mono_of_cancel_zero {Q R : C} (f : Q ⟶ R) (h : ∀ {P : C} (g : P ⟶ Q), g ≫ f = 0 → g = 0) :
    Mono f :=
  ⟨fun P g g' hg =>
    sub_eq_zero.1 <| h _ <| (map_sub (rightComp P f) g g').trans <| sub_eq_zero.2 hg⟩
#align category_theory.preadditive.mono_of_cancel_zero CategoryTheory.Preadditive.mono_of_cancel_zero
-/

#print CategoryTheory.Preadditive.mono_iff_cancel_zero /-
theorem mono_iff_cancel_zero {Q R : C} (f : Q ⟶ R) :
    Mono f ↔ ∀ (P : C) (g : P ⟶ Q), g ≫ f = 0 → g = 0 :=
  ⟨fun m P g => zero_of_comp_mono _, mono_of_cancel_zero f⟩
#align category_theory.preadditive.mono_iff_cancel_zero CategoryTheory.Preadditive.mono_iff_cancel_zero
-/

#print CategoryTheory.Preadditive.mono_of_kernel_zero /-
theorem mono_of_kernel_zero {X Y : C} {f : X ⟶ Y} [HasLimit (parallelPair f 0)]
    (w : kernel.ι f = 0) : Mono f :=
  mono_of_cancel_zero f fun P g h => by rw [← kernel.lift_ι f g h, w, limits.comp_zero]
#align category_theory.preadditive.mono_of_kernel_zero CategoryTheory.Preadditive.mono_of_kernel_zero
-/

#print CategoryTheory.Preadditive.epi_of_cancel_zero /-
theorem epi_of_cancel_zero {P Q : C} (f : P ⟶ Q) (h : ∀ {R : C} (g : Q ⟶ R), f ≫ g = 0 → g = 0) :
    Epi f :=
  ⟨fun R g g' hg => sub_eq_zero.1 <| h _ <| (map_sub (leftComp R f) g g').trans <| sub_eq_zero.2 hg⟩
#align category_theory.preadditive.epi_of_cancel_zero CategoryTheory.Preadditive.epi_of_cancel_zero
-/

#print CategoryTheory.Preadditive.epi_iff_cancel_zero /-
theorem epi_iff_cancel_zero {P Q : C} (f : P ⟶ Q) :
    Epi f ↔ ∀ (R : C) (g : Q ⟶ R), f ≫ g = 0 → g = 0 :=
  ⟨fun e R g => zero_of_epi_comp _, epi_of_cancel_zero f⟩
#align category_theory.preadditive.epi_iff_cancel_zero CategoryTheory.Preadditive.epi_iff_cancel_zero
-/

#print CategoryTheory.Preadditive.epi_of_cokernel_zero /-
theorem epi_of_cokernel_zero {X Y : C} {f : X ⟶ Y} [HasColimit (parallelPair f 0)]
    (w : cokernel.π f = 0) : Epi f :=
  epi_of_cancel_zero f fun P g h => by rw [← cokernel.π_desc f g h, w, limits.zero_comp]
#align category_theory.preadditive.epi_of_cokernel_zero CategoryTheory.Preadditive.epi_of_cokernel_zero
-/

namespace IsIso

#print CategoryTheory.Preadditive.IsIso.comp_left_eq_zero /-
@[simp]
theorem comp_left_eq_zero [IsIso f] : f ≫ g = 0 ↔ g = 0 := by
  rw [← is_iso.eq_inv_comp, limits.comp_zero]
#align category_theory.preadditive.is_iso.comp_left_eq_zero CategoryTheory.Preadditive.IsIso.comp_left_eq_zero
-/

#print CategoryTheory.Preadditive.IsIso.comp_right_eq_zero /-
@[simp]
theorem comp_right_eq_zero [IsIso g] : f ≫ g = 0 ↔ f = 0 := by
  rw [← is_iso.eq_comp_inv, limits.zero_comp]
#align category_theory.preadditive.is_iso.comp_right_eq_zero CategoryTheory.Preadditive.IsIso.comp_right_eq_zero
-/

end IsIso

open ZeroObject

variable [HasZeroObject C]

#print CategoryTheory.Preadditive.mono_of_kernel_iso_zero /-
theorem mono_of_kernel_iso_zero {X Y : C} {f : X ⟶ Y} [HasLimit (parallelPair f 0)]
    (w : kernel f ≅ 0) : Mono f :=
  mono_of_kernel_zero (zero_of_source_iso_zero _ w)
#align category_theory.preadditive.mono_of_kernel_iso_zero CategoryTheory.Preadditive.mono_of_kernel_iso_zero
-/

#print CategoryTheory.Preadditive.epi_of_cokernel_iso_zero /-
theorem epi_of_cokernel_iso_zero {X Y : C} {f : X ⟶ Y} [HasColimit (parallelPair f 0)]
    (w : cokernel f ≅ 0) : Epi f :=
  epi_of_cokernel_zero (zero_of_target_iso_zero _ w)
#align category_theory.preadditive.epi_of_cokernel_iso_zero CategoryTheory.Preadditive.epi_of_cokernel_iso_zero
-/

end Preadditive

section Equalizers

variable {C : Type u} [Category.{v} C] [Preadditive C]

section

variable {X Y : C} {f : X ⟶ Y} {g : X ⟶ Y}

/- warning: category_theory.preadditive.fork_of_kernel_fork -> CategoryTheory.Preadditive.forkOfKernelFork is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.fork_of_kernel_fork CategoryTheory.Preadditive.forkOfKernelForkₓ'. -/
/-- Map a kernel cone on the difference of two morphisms to the equalizer fork. -/
@[simps pt]
def forkOfKernelFork (c : KernelFork (f - g)) : Fork f g :=
  Fork.ofι c.ι <| by rw [← sub_eq_zero, ← comp_sub, c.condition]
#align category_theory.preadditive.fork_of_kernel_fork CategoryTheory.Preadditive.forkOfKernelFork

/- warning: category_theory.preadditive.fork_of_kernel_fork_ι -> CategoryTheory.Preadditive.forkOfKernelFork_ι is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.fork_of_kernel_fork_ι CategoryTheory.Preadditive.forkOfKernelFork_ιₓ'. -/
@[simp]
theorem forkOfKernelFork_ι (c : KernelFork (f - g)) : (forkOfKernelFork c).ι = c.ι :=
  rfl
#align category_theory.preadditive.fork_of_kernel_fork_ι CategoryTheory.Preadditive.forkOfKernelFork_ι

/- warning: category_theory.preadditive.kernel_fork_of_fork -> CategoryTheory.Preadditive.kernelForkOfFork is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.kernel_fork_of_fork CategoryTheory.Preadditive.kernelForkOfForkₓ'. -/
/-- Map any equalizer fork to a cone on the difference of the two morphisms. -/
def kernelForkOfFork (c : Fork f g) : KernelFork (f - g) :=
  Fork.ofι c.ι <| by rw [comp_sub, comp_zero, sub_eq_zero, c.condition]
#align category_theory.preadditive.kernel_fork_of_fork CategoryTheory.Preadditive.kernelForkOfFork

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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.kernel_fork_of_fork_ι CategoryTheory.Preadditive.kernelForkOfFork_ιₓ'. -/
@[simp]
theorem kernelForkOfFork_ι (c : Fork f g) : (kernelForkOfFork c).ι = c.ι :=
  rfl
#align category_theory.preadditive.kernel_fork_of_fork_ι CategoryTheory.Preadditive.kernelForkOfFork_ι

/- warning: category_theory.preadditive.kernel_fork_of_fork_of_ι -> CategoryTheory.Preadditive.kernelForkOfFork_ofι is a dubious translation:
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_inst_1)) P Y) (SubtractionMonoid.toSubNegZeroMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Y) (AddGroup.toSubtractionMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) P Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 P Y))))))))))))
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.kernel_fork_of_fork_of_ι CategoryTheory.Preadditive.kernelForkOfFork_ofιₓ'. -/
@[simp]
theorem kernelForkOfFork_ofι {P : C} (ι : P ⟶ X) (w : ι ≫ f = ι ≫ g) :
    kernelForkOfFork (Fork.ofι ι w) = KernelFork.ofι ι (by simp [w]) :=
  rfl
#align category_theory.preadditive.kernel_fork_of_fork_of_ι CategoryTheory.Preadditive.kernelForkOfFork_ofι

/- warning: category_theory.preadditive.is_limit_fork_of_kernel_fork -> CategoryTheory.Preadditive.isLimitForkOfKernelFork is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.is_limit_fork_of_kernel_fork CategoryTheory.Preadditive.isLimitForkOfKernelForkₓ'. -/
/-- A kernel of `f - g` is an equalizer of `f` and `g`. -/
def isLimitForkOfKernelFork {c : KernelFork (f - g)} (i : IsLimit c) :
    IsLimit (forkOfKernelFork c) :=
  Fork.IsLimit.mk' _ fun s =>
    ⟨i.lift (kernelForkOfFork s), i.fac _ _, fun m h => by apply fork.is_limit.hom_ext i <;> tidy⟩
#align category_theory.preadditive.is_limit_fork_of_kernel_fork CategoryTheory.Preadditive.isLimitForkOfKernelFork

/- warning: category_theory.preadditive.is_limit_fork_of_kernel_fork_lift -> CategoryTheory.Preadditive.isLimitForkOfKernelFork_lift is a dubious translation:
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_inst_1 _inst_2 X Y f g s))
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.is_limit_fork_of_kernel_fork_lift CategoryTheory.Preadditive.isLimitForkOfKernelFork_liftₓ'. -/
@[simp]
theorem isLimitForkOfKernelFork_lift {c : KernelFork (f - g)} (i : IsLimit c) (s : Fork f g) :
    (isLimitForkOfKernelFork i).lift s = i.lift (kernelForkOfFork s) :=
  rfl
#align category_theory.preadditive.is_limit_fork_of_kernel_fork_lift CategoryTheory.Preadditive.isLimitForkOfKernelFork_lift

/- warning: category_theory.preadditive.is_limit_kernel_fork_of_fork -> CategoryTheory.Preadditive.isLimitKernelForkOfFork is a dubious translation:
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  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} {f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y} {g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y} {c : CategoryTheory.Limits.Fork.{u1, u2} C _inst_1 X Y f g}, (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y f g) c) -> (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toHasSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g) (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y))))) (CategoryTheory.Preadditive.kernelForkOfFork.{u1, u2} C _inst_1 _inst_2 X Y f g c))
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} {f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y} {g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y} {c : CategoryTheory.Limits.Fork.{u1, u2} C _inst_1 X Y f g}, (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y f g) c) -> (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g) (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y)))) (CategoryTheory.Preadditive.kernelForkOfFork.{u1, u2} C _inst_1 _inst_2 X Y f g c))
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.is_limit_kernel_fork_of_fork CategoryTheory.Preadditive.isLimitKernelForkOfForkₓ'. -/
/-- An equalizer of `f` and `g` is a kernel of `f - g`. -/
def isLimitKernelForkOfFork {c : Fork f g} (i : IsLimit c) : IsLimit (kernelForkOfFork c) :=
  Fork.IsLimit.mk' _ fun s =>
    ⟨i.lift (forkOfKernelFork s), i.fac _ _, fun m h => by apply fork.is_limit.hom_ext i <;> tidy⟩
#align category_theory.preadditive.is_limit_kernel_fork_of_fork CategoryTheory.Preadditive.isLimitKernelForkOfFork

variable (f g)

/- warning: category_theory.preadditive.has_equalizer_of_has_kernel -> CategoryTheory.Preadditive.hasEqualizer_of_hasKernel is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasKernel.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toHasSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)], CategoryTheory.Limits.HasEqualizer.{u1, u2} C _inst_1 X Y f g
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasKernel.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)], CategoryTheory.Limits.HasEqualizer.{u1, u2} C _inst_1 X Y f g
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.has_equalizer_of_has_kernel CategoryTheory.Preadditive.hasEqualizer_of_hasKernelₓ'. -/
/-- A preadditive category has an equalizer for `f` and `g` if it has a kernel for `f - g`. -/
theorem hasEqualizer_of_hasKernel [HasKernel (f - g)] : HasEqualizer f g :=
  HasLimit.mk
    { Cone := forkOfKernelFork _
      IsLimit := isLimitForkOfKernelFork (equalizerIsEqualizer (f - g) 0) }
#align category_theory.preadditive.has_equalizer_of_has_kernel CategoryTheory.Preadditive.hasEqualizer_of_hasKernel

/- warning: category_theory.preadditive.has_kernel_of_has_equalizer -> CategoryTheory.Preadditive.hasKernel_of_hasEqualizer is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasEqualizer.{u1, u2} C _inst_1 X Y f g], CategoryTheory.Limits.HasKernel.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toHasSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasEqualizer.{u1, u2} C _inst_1 X Y f g], CategoryTheory.Limits.HasKernel.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.has_kernel_of_has_equalizer CategoryTheory.Preadditive.hasKernel_of_hasEqualizerₓ'. -/
/-- A preadditive category has a kernel for `f - g` if it has an equalizer for `f` and `g`. -/
theorem hasKernel_of_hasEqualizer [HasEqualizer f g] : HasKernel (f - g) :=
  HasLimit.mk
    { Cone := kernelForkOfFork (equalizer.fork f g)
      IsLimit := isLimitKernelForkOfFork (limit.isLimit (parallelPair f g)) }
#align category_theory.preadditive.has_kernel_of_has_equalizer CategoryTheory.Preadditive.hasKernel_of_hasEqualizer

variable {f g}

/- warning: category_theory.preadditive.cofork_of_cokernel_cofork -> CategoryTheory.Preadditive.coforkOfCokernelCofork is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} {f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y} {g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y}, (CategoryTheory.Limits.CokernelCofork.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toHasSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)) -> (CategoryTheory.Limits.Cofork.{u1, u2} C _inst_1 X Y f g)
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.cofork_of_cokernel_cofork CategoryTheory.Preadditive.coforkOfCokernelCoforkₓ'. -/
/-- Map a cokernel cocone on the difference of two morphisms to the coequalizer cofork. -/
@[simps pt]
def coforkOfCokernelCofork (c : CokernelCofork (f - g)) : Cofork f g :=
  Cofork.ofπ c.π <| by rw [← sub_eq_zero, ← sub_comp, c.condition]
#align category_theory.preadditive.cofork_of_cokernel_cofork CategoryTheory.Preadditive.coforkOfCokernelCofork

/- warning: category_theory.preadditive.cofork_of_cokernel_cofork_π -> CategoryTheory.Preadditive.coforkOfCokernelCofork_π is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.cofork_of_cokernel_cofork_π CategoryTheory.Preadditive.coforkOfCokernelCofork_πₓ'. -/
@[simp]
theorem coforkOfCokernelCofork_π (c : CokernelCofork (f - g)) :
    (coforkOfCokernelCofork c).π = c.π :=
  rfl
#align category_theory.preadditive.cofork_of_cokernel_cofork_π CategoryTheory.Preadditive.coforkOfCokernelCofork_π

/- warning: category_theory.preadditive.cokernel_cofork_of_cofork -> CategoryTheory.Preadditive.cokernelCoforkOfCofork is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.cokernel_cofork_of_cofork CategoryTheory.Preadditive.cokernelCoforkOfCoforkₓ'. -/
/-- Map any coequalizer cofork to a cocone on the difference of the two morphisms. -/
def cokernelCoforkOfCofork (c : Cofork f g) : CokernelCofork (f - g) :=
  Cofork.ofπ c.π <| by rw [sub_comp, zero_comp, sub_eq_zero, c.condition]
#align category_theory.preadditive.cokernel_cofork_of_cofork CategoryTheory.Preadditive.cokernelCoforkOfCofork

/- warning: category_theory.preadditive.cokernel_cofork_of_cofork_π -> CategoryTheory.Preadditive.cokernelCoforkOfCofork_π is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.cokernel_cofork_of_cofork_π CategoryTheory.Preadditive.cokernelCoforkOfCofork_πₓ'. -/
@[simp]
theorem cokernelCoforkOfCofork_π (c : Cofork f g) : (cokernelCoforkOfCofork c).π = c.π :=
  rfl
#align category_theory.preadditive.cokernel_cofork_of_cofork_π CategoryTheory.Preadditive.cokernelCoforkOfCofork_π

/- warning: category_theory.preadditive.cokernel_cofork_of_cofork_of_π -> CategoryTheory.Preadditive.cokernelCoforkOfCofork_ofπ is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.cokernel_cofork_of_cofork_of_π CategoryTheory.Preadditive.cokernelCoforkOfCofork_ofπₓ'. -/
@[simp]
theorem cokernelCoforkOfCofork_ofπ {P : C} (π : Y ⟶ P) (w : f ≫ π = g ≫ π) :
    cokernelCoforkOfCofork (Cofork.ofπ π w) = CokernelCofork.ofπ π (by simp [w]) :=
  rfl
#align category_theory.preadditive.cokernel_cofork_of_cofork_of_π CategoryTheory.Preadditive.cokernelCoforkOfCofork_ofπ

/- warning: category_theory.preadditive.is_colimit_cofork_of_cokernel_cofork -> CategoryTheory.Preadditive.isColimitCoforkOfCokernelCofork is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.is_colimit_cofork_of_cokernel_cofork CategoryTheory.Preadditive.isColimitCoforkOfCokernelCoforkₓ'. -/
/-- A cokernel of `f - g` is a coequalizer of `f` and `g`. -/
def isColimitCoforkOfCokernelCofork {c : CokernelCofork (f - g)} (i : IsColimit c) :
    IsColimit (coforkOfCokernelCofork c) :=
  Cofork.IsColimit.mk' _ fun s =>
    ⟨i.desc (cokernelCoforkOfCofork s), i.fac _ _, fun m h => by
      apply cofork.is_colimit.hom_ext i <;> tidy⟩
#align category_theory.preadditive.is_colimit_cofork_of_cokernel_cofork CategoryTheory.Preadditive.isColimitCoforkOfCokernelCofork

/- warning: category_theory.preadditive.is_colimit_cofork_of_cokernel_cofork_desc -> CategoryTheory.Preadditive.isColimitCoforkOfCokernelCofork_desc is a dubious translation:
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(CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)} (i : CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g) (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y)))) c) (s : CategoryTheory.Limits.Cofork.{u1, u2} C _inst_1 X Y f g), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y f g) (CategoryTheory.Preadditive.coforkOfCokernelCofork.{u1, u2} C _inst_1 _inst_2 X Y f g c)) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y f g) s)) (CategoryTheory.Limits.IsColimit.desc.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y f g) (CategoryTheory.Preadditive.coforkOfCokernelCofork.{u1, u2} C _inst_1 _inst_2 X Y f g c) (CategoryTheory.Preadditive.isColimitCoforkOfCokernelCofork.{u1, u2} C _inst_1 _inst_2 X Y f g c i) s) (CategoryTheory.Limits.IsColimit.desc.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g) (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y)))) c i (CategoryTheory.Preadditive.cokernelCoforkOfCofork.{u1, u2} C _inst_1 _inst_2 X Y f g s))
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.is_colimit_cofork_of_cokernel_cofork_desc CategoryTheory.Preadditive.isColimitCoforkOfCokernelCofork_descₓ'. -/
@[simp]
theorem isColimitCoforkOfCokernelCofork_desc {c : CokernelCofork (f - g)} (i : IsColimit c)
    (s : Cofork f g) :
    (isColimitCoforkOfCokernelCofork i).desc s = i.desc (cokernelCoforkOfCofork s) :=
  rfl
#align category_theory.preadditive.is_colimit_cofork_of_cokernel_cofork_desc CategoryTheory.Preadditive.isColimitCoforkOfCokernelCofork_desc

/- warning: category_theory.preadditive.is_colimit_cokernel_cofork_of_cofork -> CategoryTheory.Preadditive.isColimitCokernelCoforkOfCofork is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} {f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y} {g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y} {c : CategoryTheory.Limits.Cofork.{u1, u2} C _inst_1 X Y f g}, (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y f g) c) -> (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toHasSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g) (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) 0 (OfNat.mk.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) 0 (Zero.zero.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.hasZero.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y))))) (CategoryTheory.Preadditive.cokernelCoforkOfCofork.{u1, u2} C _inst_1 _inst_2 X Y f g c))
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} {f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y} {g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y} {c : CategoryTheory.Limits.Cofork.{u1, u2} C _inst_1 X Y f g}, (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y f g) c) -> (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C _inst_1 (CategoryTheory.Limits.parallelPair.{u1, u2} C _inst_1 X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g) (OfNat.ofNat.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) 0 (Zero.toOfNat0.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y)))) (CategoryTheory.Preadditive.cokernelCoforkOfCofork.{u1, u2} C _inst_1 _inst_2 X Y f g c))
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.is_colimit_cokernel_cofork_of_cofork CategoryTheory.Preadditive.isColimitCokernelCoforkOfCoforkₓ'. -/
/-- A coequalizer of `f` and `g` is a cokernel of `f - g`. -/
def isColimitCokernelCoforkOfCofork {c : Cofork f g} (i : IsColimit c) :
    IsColimit (cokernelCoforkOfCofork c) :=
  Cofork.IsColimit.mk' _ fun s =>
    ⟨i.desc (coforkOfCokernelCofork s), i.fac _ _, fun m h => by
      apply cofork.is_colimit.hom_ext i <;> tidy⟩
#align category_theory.preadditive.is_colimit_cokernel_cofork_of_cofork CategoryTheory.Preadditive.isColimitCokernelCoforkOfCofork

variable (f g)

/- warning: category_theory.preadditive.has_coequalizer_of_has_cokernel -> CategoryTheory.Preadditive.hasCoequalizer_of_hasCokernel is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCokernel.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toHasSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)], CategoryTheory.Limits.HasCoequalizer.{u1, u2} C _inst_1 X Y f g
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCokernel.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)], CategoryTheory.Limits.HasCoequalizer.{u1, u2} C _inst_1 X Y f g
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.has_coequalizer_of_has_cokernel CategoryTheory.Preadditive.hasCoequalizer_of_hasCokernelₓ'. -/
/-- A preadditive category has a coequalizer for `f` and `g` if it has a cokernel for `f - g`. -/
theorem hasCoequalizer_of_hasCokernel [HasCokernel (f - g)] : HasCoequalizer f g :=
  HasColimit.mk
    { Cocone := coforkOfCokernelCofork _
      IsColimit := isColimitCoforkOfCokernelCofork (coequalizerIsCoequalizer (f - g) 0) }
#align category_theory.preadditive.has_coequalizer_of_has_cokernel CategoryTheory.Preadditive.hasCoequalizer_of_hasCokernel

/- warning: category_theory.preadditive.has_cokernel_of_has_coequalizer -> CategoryTheory.Preadditive.hasCokernel_of_hasCoequalizer is a dubious translation:
lean 3 declaration is
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u2} C _inst_1 X Y f g], CategoryTheory.Limits.HasCokernel.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toHasSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)
but is expected to have type
  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Preadditive.{u1, u2} C _inst_1] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (g : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.HasCoequalizer.{u1, u2} C _inst_1 X Y f g], CategoryTheory.Limits.HasCokernel.{u1, u2} C _inst_1 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u1, u2} C _inst_1 _inst_2) X Y (HSub.hSub.{u1, u1, u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (instHSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (SubNegMonoid.toSub.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddGroup.toSubNegMonoid.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (AddCommGroup.toAddGroup.{u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) X Y) (CategoryTheory.Preadditive.homGroup.{u1, u2} C _inst_1 _inst_2 X Y))))) f g)
Case conversion may be inaccurate. Consider using '#align category_theory.preadditive.has_cokernel_of_has_coequalizer CategoryTheory.Preadditive.hasCokernel_of_hasCoequalizerₓ'. -/
/-- A preadditive category has a cokernel for `f - g` if it has a coequalizer for `f` and `g`. -/
theorem hasCokernel_of_hasCoequalizer [HasCoequalizer f g] : HasCokernel (f - g) :=
  HasColimit.mk
    { Cocone := cokernelCoforkOfCofork (coequalizer.cofork f g)
      IsColimit := isColimitCokernelCoforkOfCofork (colimit.isColimit (parallelPair f g)) }
#align category_theory.preadditive.has_cokernel_of_has_coequalizer CategoryTheory.Preadditive.hasCokernel_of_hasCoequalizer

end

#print CategoryTheory.Preadditive.hasEqualizers_of_hasKernels /-
/-- If a preadditive category has all kernels, then it also has all equalizers. -/
theorem hasEqualizers_of_hasKernels [HasKernels C] : HasEqualizers C :=
  @hasEqualizers_of_hasLimit_parallelPair _ _ fun _ _ f g => hasEqualizer_of_hasKernel f g
#align category_theory.preadditive.has_equalizers_of_has_kernels CategoryTheory.Preadditive.hasEqualizers_of_hasKernels
-/

#print CategoryTheory.Preadditive.hasCoequalizers_of_hasCokernels /-
/-- If a preadditive category has all cokernels, then it also has all coequalizers. -/
theorem hasCoequalizers_of_hasCokernels [HasCokernels C] : HasCoequalizers C :=
  @hasCoequalizers_of_hasColimit_parallelPair _ _ fun _ _ f g => hasCoequalizer_of_hasCokernel f g
#align category_theory.preadditive.has_coequalizers_of_has_cokernels CategoryTheory.Preadditive.hasCoequalizers_of_hasCokernels
-/

end Equalizers

end Preadditive

end CategoryTheory