-- 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
P : PresheafOfModules.{v, v', u', u} R₀
F : SheafOfModules.{v, v', u', u} R
f : (sheafification α).obj P ⟶ F
a :
  Eq (α := P.presheaf ⟶ Functor.obj (sheafToPresheaf J AddCommGrpCat) ((SheafOfModules.toSheaf R).obj F))
    (CategoryStruct.comp (Y :=
        Functor.obj (presheafToSheaf J AddCommGrpCat ⋙ sheafToPresheaf J AddCommGrpCat) P.presheaf)
        (NatTrans.app (Adjunction.unit (sheafificationAdjunction J AddCommGrpCat)) P.presheaf)
        (CategoryTheory.Functor.map (sheafToPresheaf J AddCommGrpCat) ((SheafOfModules.toSheaf R).map f)) :
      P.presheaf ⟶ Functor.obj (sheafToPresheaf J AddCommGrpCat) ((SheafOfModules.toSheaf R).obj F))
    ((toPresheaf R₀).map
        ((sheafificationHomEquiv α :
            (a : (sheafification α).obj P ⟶ F) → P ⟶ (restrictScalars α).obj ((SheafOfModules.forget R).obj F))
          f) :
      (toPresheaf R₀).obj P ⟶ (toPresheaf R₀).obj ((restrictScalars α).obj ((SheafOfModules.forget R).obj F)))
⊢ Eq (α := P.presheaf ⟶ Functor.obj (sheafToPresheaf J AddCommGrpCat) ((SheafOfModules.toSheaf R).obj F))
    ((Adjunction.homEquiv (sheafificationAdjunction J AddCommGrpCat) P.presheaf ((SheafOfModules.toSheaf R).obj F) :
          (a : (presheafToSheaf J AddCommGrpCat).obj P.presheaf ⟶ (SheafOfModules.toSheaf R).obj F) →
            P.presheaf ⟶ (sheafToPresheaf J AddCommGrpCat).obj ((SheafOfModules.toSheaf R).obj F))
        ((SheafOfModules.toSheaf R).map
          ((Equiv.symm (sheafificationHomEquiv α) :
              (a : P ⟶ (restrictScalars α).obj ((SheafOfModules.forget R).obj F)) → (sheafification α).obj P ⟶ F)
            ((sheafificationHomEquiv α :
                (a : (sheafification α).obj P ⟶ F) → P ⟶ (restrictScalars α).obj ((SheafOfModules.forget R).obj F))
              f))) :
      P.presheaf ⟶ Functor.obj (sheafToPresheaf J AddCommGrpCat) ((SheafOfModules.toSheaf R).obj F))
    ((Adjunction.homEquiv (sheafificationAdjunction J AddCommGrpCat) P.presheaf ((SheafOfModules.toSheaf R).obj F) :
          (a : (presheafToSheaf J AddCommGrpCat).obj P.presheaf ⟶ (SheafOfModules.toSheaf R).obj F) →
            P.presheaf ⟶ (sheafToPresheaf J AddCommGrpCat).obj ((SheafOfModules.toSheaf R).obj F))
        ((Equiv.symm
              (Adjunction.homEquiv (sheafificationAdjunction J AddCommGrpCat) P.presheaf
                ((SheafOfModules.toSheaf R).obj F)) :
            (a : P.presheaf ⟶ (sheafToPresheaf J AddCommGrpCat).obj ((SheafOfModules.toSheaf R).obj F)) →
              (presheafToSheaf J AddCommGrpCat).obj P.presheaf ⟶ (SheafOfModules.toSheaf R).obj F)
          ((toPresheaf R₀).map
            ((sheafificationHomEquiv α :
                (a : (sheafification α).obj P ⟶ F) → P ⟶ (restrictScalars α).obj ((SheafOfModules.forget R).obj F))
              f))) :
      P.presheaf ⟶ Functor.obj (sheafToPresheaf J AddCommGrpCat) ((SheafOfModules.toSheaf R).obj F))

Tactic:
rw [Equiv.apply_symm_apply, Adjunction.homEquiv_unit, Equiv.symm_apply_apply]

Premises used: Equiv.apply_symm_apply, Equiv.symm_apply_apply, CategoryTheory.Adjunction.homEquiv_unit