-- Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
D_1 : Type u₂
inst_1 : Category.{v₂, u₂} D_1
D' : Type u₃
inst_2 : Category.{v₃, u₃} D'
D'' : Type u₄
inst_3 : Category.{v₄, u₄} D''
J_1 : GrothendieckTopology.{v₁, u₁} C_1
K_1 : GrothendieckTopology.{v₂, u₂} D_1
F_1 : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C_1 D_1
S_1 : Sheaf J_1 RingCat
R : Sheaf K_1 RingCat
inst_4 : F_1.IsContinuous J_1 K_1
φ_1 : S_1 ⟶ obj (F_1.sheafPushforwardContinuous RingCat J_1 K_1) R
C : Type u₁
inst_5 : Category.{v₁, u₁} C
D : Type u₂
inst_6 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F G H : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
T : Sheaf J RingCat
S : Sheaf K RingCat
inst_7 : F.IsContinuous J K
inst_8 : G.IsContinuous J K
inst_9 : H.IsContinuous J K
φ : T ⟶ obj (G.sheafPushforwardContinuous RingCat J K) S
α : F ⟶ G
X : SheafOfModules.{v, v₂, u₂, u} S
U V : Cᵒᵖ
i : U ⟶ V
hf :
  ∀ (x : (↑(((pushforward φ).obj X).val.obj U) : Type v)),
    Eq (α :=
      (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (T.obj (P := Presheaf.IsSheaf J).map i))).obj
            (((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val.obj V)) :
        Type v))
      ((ModuleCat.Hom.hom
              (((pushforward φ).obj X).val.map i ≫
                (ModuleCat.restrictScalars (RingCat.Hom.hom (T.obj (P := Presheaf.IsSheaf J).map i))).map
                  ((ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app V))).map
                    (X.val.map (Quiver.Hom.op (α.app (Opposite.unop V)))))) :
            (a : (↑(((pushforward φ).obj X).val.obj U) : Type v)) →
              (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (T.obj.map i))).obj
                    (((pushforward (φ ≫ (sheafPushforwardContinuousNatTrans α RingCat J K).app S)).obj X).val.obj V)) :
                Type v))
          x :
        (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (T.obj (P := Presheaf.IsSheaf J).map i))).obj
              (((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val.obj
                V)) :
          Type v))
      ((ModuleCat.Hom.hom
              ((ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app U))).map
                  (X.val.map (Quiver.Hom.op (α.app (Opposite.unop U)))) ≫
                ((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val.map
                  i) :
            (a : (↑(((pushforward φ).obj X).val.obj U) : Type v)) →
              (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (T.obj.map i))).obj
                    (((pushforward (φ ≫ (sheafPushforwardContinuousNatTrans α RingCat J K).app S)).obj X).val.obj V)) :
                Type v))
          x :
        (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (T.obj (P := Presheaf.IsSheaf J).map i))).obj
              (((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val.obj
                V)) :
          Type v))
⊢ ((pushforward φ).obj X).val.map i ≫
      (ModuleCat.restrictScalars (RingCat.Hom.hom (T.obj (P := Presheaf.IsSheaf J).map i))).map
        ((ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app V))).map
          (X.val.map (Quiver.Hom.op (α.app (Opposite.unop V))))) =
    (ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app U))).map
        (X.val.map (Quiver.Hom.op (α.app (Opposite.unop U)))) ≫
      ((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val.map i

Tactic:
ext x