-- 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
M : SheafOfModules.{v, v', u', u} R
rw :
  Eq (α :=
    (SheafOfModules.toSheaf R).obj ((sheafification α).obj ((SheafOfModules.forget R ⋙ restrictScalars α).obj M)) ⟶
      (SheafOfModules.toSheaf R).obj (Functor.obj (𝟭 (SheafOfModules.{v, v', u', u} R)) M))
    ((Equiv.symm
            (Adjunction.homEquiv (CategoryTheory.sheafificationAdjunction J AddCommGrpCat)
              ((SheafOfModules.forget R ⋙ restrictScalars α).obj M).presheaf
              ((SheafOfModules.toSheaf R).obj (Functor.obj (𝟭 (SheafOfModules.{v, v', u', u} R)) M))) :
          (a :
              ((SheafOfModules.forget R ⋙ restrictScalars α).obj M).presheaf ⟶
                (sheafToPresheaf J AddCommGrpCat).obj ((SheafOfModules.toSheaf R).obj ((𝟭 (SheafOfModules R)).obj M))) →
            (presheafToSheaf J AddCommGrpCat).obj ((SheafOfModules.forget R ⋙ restrictScalars α).obj M).presheaf ⟶
              (SheafOfModules.toSheaf R).obj ((𝟭 (SheafOfModules R)).obj M))
        ((toPresheaf R₀).map (𝟙 ((SheafOfModules.forget R ⋙ restrictScalars α).obj M))) :
      Functor.obj (presheafToSheaf J AddCommGrpCat) ((SheafOfModules.forget R ⋙ restrictScalars α).obj M).presheaf ⟶
        (SheafOfModules.toSheaf R).obj (Functor.obj (𝟭 (SheafOfModules.{v, v', u', u} R)) M))
    ((Equiv.symm
            (Adjunction.homEquiv (CategoryTheory.sheafificationAdjunction J AddCommGrpCat)
              (Functor.obj (sheafToPresheaf J AddCommGrpCat) ((SheafOfModules.toSheaf R).obj M))
              ((SheafOfModules.toSheaf R).obj M)) :
          (a :
              (sheafToPresheaf J AddCommGrpCat).obj ((SheafOfModules.toSheaf R).obj M) ⟶
                (sheafToPresheaf J AddCommGrpCat).obj ((SheafOfModules.toSheaf R).obj M)) →
            (presheafToSheaf J AddCommGrpCat).obj
                ((sheafToPresheaf J AddCommGrpCat).obj ((SheafOfModules.toSheaf R).obj M)) ⟶
              (SheafOfModules.toSheaf R).obj M)
        (𝟙 (Functor.obj (sheafToPresheaf J AddCommGrpCat) ((SheafOfModules.toSheaf R).obj M))) :
      Functor.obj (presheafToSheaf J AddCommGrpCat)
          (Functor.obj (sheafToPresheaf J AddCommGrpCat) ((SheafOfModules.toSheaf R).obj M)) ⟶
        (SheafOfModules.toSheaf R).obj M)
⊢ Eq (α :=
    (SheafOfModules.toSheaf R).obj ((sheafification α).obj ((SheafOfModules.forget R ⋙ restrictScalars α).obj M)) ⟶
      (SheafOfModules.toSheaf R).obj (Functor.obj (𝟭 (SheafOfModules.{v, v', u', u} R)) M))
    ((Equiv.symm
            (Adjunction.homEquiv (CategoryTheory.sheafificationAdjunction J AddCommGrpCat)
              ((SheafOfModules.forget R ⋙ restrictScalars α).obj M).presheaf
              ((SheafOfModules.toSheaf R).obj (Functor.obj (𝟭 (SheafOfModules.{v, v', u', u} R)) M))) :
          (a :
              ((SheafOfModules.forget R ⋙ restrictScalars α).obj M).presheaf ⟶
                (sheafToPresheaf J AddCommGrpCat).obj ((SheafOfModules.toSheaf R).obj ((𝟭 (SheafOfModules R)).obj M))) →
            (presheafToSheaf J AddCommGrpCat).obj ((SheafOfModules.forget R ⋙ restrictScalars α).obj M).presheaf ⟶
              (SheafOfModules.toSheaf R).obj ((𝟭 (SheafOfModules R)).obj M))
        ((toPresheaf R₀).map (𝟙 ((SheafOfModules.forget R ⋙ restrictScalars α).obj M))) :
      Functor.obj (presheafToSheaf J AddCommGrpCat) ((SheafOfModules.forget R ⋙ restrictScalars α).obj M).presheaf ⟶
        (SheafOfModules.toSheaf R).obj (Functor.obj (𝟭 (SheafOfModules.{v, v', u', u} R)) M))
    (NatTrans.app (Adjunction.counit (CategoryTheory.sheafificationAdjunction J AddCommGrpCat))
        ((SheafOfModules.toSheaf R).obj M) :
      Functor.obj (sheafToPresheaf J AddCommGrpCat ⋙ presheafToSheaf J AddCommGrpCat)
          ((SheafOfModules.toSheaf R).obj M) ⟶
        Functor.obj (𝟭 (Sheaf J AddCommGrpCat)) ((SheafOfModules.toSheaf R).obj M))

Tactic:
rw [← Adjunction.homEquiv_symm_id]

Premises used: CategoryTheory.Adjunction.homEquiv_symm_id