-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification
Goal:
C : Type u'
inst : Category.{v', u'} C
J : GrothendieckTopology.{v', u'} C
R₀ : CategoryTheory.Functor.{v', u, u', u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
inst_3 :
  J.WEqualsLocallyBijective AddCommGrpCat (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v))
    (CA := AddCommGrpCat.carrier)
inst_4 : HasWeakSheafify J AddCommGrpCat
X Y Z : PresheafOfModules.{v, v', u', u} R₀
x : X ⟶ Y
x_1 : Y ⟶ Z
h :
  Eq (α :=
    (toPresheaf (R.obj (P := Presheaf.IsSheaf J))).obj
        (sheafify α (A := Functor.obj (presheafToSheaf J Ab) X.presheaf) (CategoryTheory.toSheafify J X.presheaf)).val ⟶
      (toPresheaf (R.obj (P := Presheaf.IsSheaf J))).obj
        (sheafify α (A := Functor.obj (presheafToSheaf J Ab) Z.presheaf) (CategoryTheory.toSheafify J Z.presheaf)).val)
    ((toPresheaf (R.obj (P := Presheaf.IsSheaf J))).map
      (homMk
        (InducedCategory.Hom.hom
            (CategoryTheory.Functor.map (presheafToSheaf J AddCommGrpCat) ((toPresheaf R₀).map x)) ≫
          InducedCategory.Hom.hom
            (CategoryTheory.Functor.map (presheafToSheaf J AddCommGrpCat) ((toPresheaf R₀).map x_1)))
        ⋯))
    (CategoryStruct.comp (Y :=
      (toPresheaf (R.obj (P := Presheaf.IsSheaf J))).obj
        (sheafify α (A := Functor.obj (presheafToSheaf J Ab) Y.presheaf) (CategoryTheory.toSheafify J Y.presheaf)).val)
      ((toPresheaf (R.obj (P := Presheaf.IsSheaf J))).map
        (homMk
          (InducedCategory.Hom.hom
            (CategoryTheory.Functor.map (presheafToSheaf J AddCommGrpCat) ((toPresheaf R₀).map x)))
          ⋯))
      ((toPresheaf (R.obj (P := Presheaf.IsSheaf J))).map
        (homMk
          (InducedCategory.Hom.hom
            (CategoryTheory.Functor.map (presheafToSheaf J AddCommGrpCat) ((toPresheaf R₀).map x_1)))
          ⋯)))
⊢ Eq (α :=
    (toPresheaf (R.obj (P := Presheaf.IsSheaf J))).obj
        (sheafify α (A := Functor.obj (presheafToSheaf J Ab) X.presheaf) (CategoryTheory.toSheafify J X.presheaf)).val ⟶
      (toPresheaf (R.obj (P := Presheaf.IsSheaf J))).obj
        (sheafify α (A := Functor.obj (presheafToSheaf J Ab) Z.presheaf) (CategoryTheory.toSheafify J Z.presheaf)).val)
    ((toPresheaf (R.obj (P := Presheaf.IsSheaf J))).map
      (SheafOfModules.Hom.val
        (sheafifyMap α (CategoryTheory.toSheafify J X.presheaf) (CategoryTheory.toSheafify J Z.presheaf) (x ≫ x_1)
          ((toPresheaf R₀ ⋙ presheafToSheaf J AddCommGrpCat).map (x ≫ x_1)) ⋯)))
    ((toPresheaf (R.obj (P := Presheaf.IsSheaf J))).map
      (SheafOfModules.Hom.val
        (CategoryStruct.comp (Y :=
          sheafify α (A := Functor.obj (presheafToSheaf J Ab) Y.presheaf) (CategoryTheory.toSheafify J Y.presheaf))
          (sheafifyMap α (CategoryTheory.toSheafify J X.presheaf) (CategoryTheory.toSheafify J Y.presheaf) x
            ((toPresheaf R₀ ⋙ presheafToSheaf J AddCommGrpCat).map x) ⋯)
          (sheafifyMap α (CategoryTheory.toSheafify J Y.presheaf) (CategoryTheory.toSheafify J Z.presheaf) x_1
            ((toPresheaf R₀ ⋙ presheafToSheaf J AddCommGrpCat).map x_1) ⋯))))

Tactic:
simp