-- Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
D' : Type u₃
inst_2 : Category.{v₃, u₃} D'
D'' : Type u₄
inst_3 : Category.{v₄, u₄} D''
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_4 : F.IsContinuous J K
φ : S ⟶ obj (F.sheafPushforwardContinuous RingCat J K) R
inst_5 : (PresheafOfModules.pushforward φ.hom).IsRightAdjoint
inst_6 : HasWeakSheafify K AddCommGrpCat
inst_7 :
  K.WEqualsLocallyBijective AddCommGrpCat (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v))
    (CA := AddCommGrpCat.carrier)
tauto tauto₁ tauto₂ tauto₃ tauto₄ tauto₅ tauto₆ tauto₇ tauto₈ tauto₉ tauto₁₀ tauto₁₁ tauto₁₂ :
  ∀ {X : SheafOfModules.{v, v₁, u₁, u} S} {Y Y' : SheafOfModules.{v, v₂, u₂, u} R}
    (f :
      (forget S ⋙
              PresheafOfModules.pullback φ.hom ⋙
                PresheafOfModules.sheafification (𝟙 (R.obj (P := Presheaf.IsSheaf K)))).obj
          X ⟶
        Y)
    (g : Y ⟶ Y'),
    Eq (α := X ⟶ (pushforward φ).obj Y')
      (((Equiv.trans
                (Adjunction.homEquiv (PresheafOfModules.sheafificationAdjunction (𝟙 (R.obj (P := Presheaf.IsSheaf K))))
                  ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) Y')
                (Equiv.trans (Adjunction.homEquiv (PresheafOfModules.pullbackPushforwardAdjunction φ.hom) X.val Y'.val)
                  (Equiv.symm (fullyFaithfulForget S).homEquiv)) :
              ((PresheafOfModules.sheafification (𝟙 (R.obj (P := Presheaf.IsSheaf K)))).obj
                    ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) ⟶
                  Y') ≃
                (X ⟶ (pushforward φ).obj Y')) :
            (a :
                (forget S ⋙ PresheafOfModules.pullback φ.hom ⋙ PresheafOfModules.sheafification (𝟙 R.obj)).obj X ⟶ Y') →
              X ⟶ (pushforward φ).obj Y')
          (f ≫ g) :
        X ⟶ (pushforward φ).obj Y')
      (((Equiv.trans
                  (Adjunction.homEquiv
                    (PresheafOfModules.sheafificationAdjunction (𝟙 (R.obj (P := Presheaf.IsSheaf K))))
                    ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) Y)
                  (Equiv.trans (Adjunction.homEquiv (PresheafOfModules.pullbackPushforwardAdjunction φ.hom) X.val Y.val)
                    (Equiv.symm (fullyFaithfulForget S).homEquiv)) :
                ((PresheafOfModules.sheafification (𝟙 (R.obj (P := Presheaf.IsSheaf K)))).obj
                      ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) ⟶
                    Y) ≃
                  (X ⟶ (pushforward φ).obj Y)) :
              (a :
                  (forget S ⋙ PresheafOfModules.pullback φ.hom ⋙ PresheafOfModules.sheafification (𝟙 R.obj)).obj X ⟶
                    Y) →
                X ⟶ (pushforward φ).obj Y)
            f ≫
          (pushforward φ).map g :
        X ⟶ (pushforward φ).obj Y')
⊢ ∀ {X : SheafOfModules.{v, v₁, u₁, u} S} {Y Y' : SheafOfModules.{v, v₂, u₂, u} R}
    (f :
      (forget S ⋙
              PresheafOfModules.pullback φ.hom ⋙
                PresheafOfModules.sheafification (𝟙 (R.obj (P := Presheaf.IsSheaf K)))).obj
          X ⟶
        Y)
    (g : Y ⟶ Y'),
    Eq (α := X ⟶ (pushforward φ).obj Y')
      (((Equiv.trans
                (Adjunction.homEquiv (PresheafOfModules.sheafificationAdjunction (𝟙 (R.obj (P := Presheaf.IsSheaf K))))
                  ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) Y')
                (Equiv.trans (Adjunction.homEquiv (PresheafOfModules.pullbackPushforwardAdjunction φ.hom) X.val Y'.val)
                  (Equiv.symm (fullyFaithfulForget S).homEquiv)) :
              ((PresheafOfModules.sheafification (𝟙 (R.obj (P := Presheaf.IsSheaf K)))).obj
                    ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) ⟶
                  Y') ≃
                (X ⟶ (pushforward φ).obj Y')) :
            (a :
                (forget S ⋙ PresheafOfModules.pullback φ.hom ⋙ PresheafOfModules.sheafification (𝟙 R.obj)).obj X ⟶ Y') →
              X ⟶ (pushforward φ).obj Y')
          (f ≫ g) :
        X ⟶ (pushforward φ).obj Y')
      (((Equiv.trans
                  (Adjunction.homEquiv
                    (PresheafOfModules.sheafificationAdjunction (𝟙 (R.obj (P := Presheaf.IsSheaf K))))
                    ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) Y)
                  (Equiv.trans (Adjunction.homEquiv (PresheafOfModules.pullbackPushforwardAdjunction φ.hom) X.val Y.val)
                    (Equiv.symm (fullyFaithfulForget S).homEquiv)) :
                ((PresheafOfModules.sheafification (𝟙 (R.obj (P := Presheaf.IsSheaf K)))).obj
                      ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) ⟶
                    Y) ≃
                  (X ⟶ (pushforward φ).obj Y)) :
              (a :
                  (forget S ⋙ PresheafOfModules.pullback φ.hom ⋙ PresheafOfModules.sheafification (𝟙 R.obj)).obj X ⟶
                    Y) →
                X ⟶ (pushforward φ).obj Y)
            f ≫
          (pushforward φ).map g :
        X ⟶ (pushforward φ).obj Y')

Tactic:
tauto