-- Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} 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_2 : F.IsContinuous J K
φ : S ⟶ Functor.obj (F.sheafPushforwardContinuous RingCat J K) R
inst_3 :
  J.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_4 :
  K.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
X Y : Cᵒᵖ
f : X ⟶ Y
⊢ Eq (α :=
    (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).obj
          (((pushforward φ).obj (unit R)).val.obj Y)) :
      Type u))
    ((ModuleCat.Hom.hom
            ((unit S).val.map f ≫
              (ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).map
                (ModuleCat.homMk
                  (CategoryTheory.Functor.map
                    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                      RingCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦
                      (↑x1 : Type u) →+ (↑x2 : Type u)) (CD := AddCommGrpCat.carrier))
                    (φ.hom.app Y))
                  ⋯)) :
          (a : (↑(S.obj.obj X) : Type u)) →
            (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj.map f))).obj
                  (((pushforward φ).obj (unit R)).val.obj Y)) :
              Type u))
        1 :
      (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).obj
            (((pushforward φ).obj (unit R)).val.obj Y)) :
        Type u))
    (((ModuleCat.Hom.hom
              (ModuleCat.homMk
                  (CategoryTheory.Functor.map
                    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                      RingCat.carrier) AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦
                      (↑x1 : Type u) →+ (↑x2 : Type u)) (CD := AddCommGrpCat.carrier))
                    (φ.hom.app X))
                  ⋯ ≫
                ((pushforward φ).obj (unit R)).val.map f) :
            LinearMap (R := (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u))
              (RingHom.id (↑(S.obj (P := Presheaf.IsSheaf J).obj X) : Type u)) (↑((unit S).val.obj X) : Type u)
              (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).obj
                    (((pushforward φ).obj (unit R)).val.obj Y)) :
                Type u)) :
          (a : (↑(S.obj.obj X) : Type u)) →
            (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj.map f))).obj
                  (((pushforward φ).obj (unit R)).val.obj Y)) :
              Type u))
        1 :
      (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (S.obj (P := Presheaf.IsSheaf J).map f))).obj
            (((pushforward φ).obj (unit R)).val.obj Y)) :
        Type u))

Tactic:
exact ConcreteCategory.congr_hom (φ.hom.naturality f) _

Premises used: CategoryTheory.NatTrans.naturality, CategoryTheory.ConcreteCategory.congr_hom, CategoryTheory.InducedCategory.Hom.hom