-- 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 Y : SheafOfModules.{v, v₂, u₂, u} S
f : X ⟶ Y
h :
  ∀ (U : Cᵒᵖ) (x : (↑(((pushforward φ).obj X).val.obj U) : Type v)),
    Eq (α :=
      (↑(((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj Y).val.obj U) :
        Type v))
      ((ModuleCat.Hom.hom
              (PresheafOfModules.Hom.app
                (Hom.val
                  ((pushforward φ).map f ≫
                    {
                      val :=
                        {
                          app := fun (U : Cᵒᵖ) ↦
                            (ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app U))).map
                              (Y.val.map (Quiver.Hom.op (α.app (Opposite.unop U)))),
                          naturality := ⋯ } }))
                U) :
            (a : (↑(((pushforward φ).obj X).val.obj U) : Type v)) →
              (↑(((pushforward (φ ≫ (sheafPushforwardContinuousNatTrans α RingCat J K).app S)).obj Y).val.obj U) :
                Type v))
          x :
        (↑(((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj Y).val.obj U) :
          Type v))
      ((ModuleCat.Hom.hom
              (PresheafOfModules.Hom.app
                (Hom.val
                  ({
                      val :=
                        {
                          app := fun (U : Cᵒᵖ) ↦
                            (ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app U))).map
                              (X.val.map (Quiver.Hom.op (α.app (Opposite.unop U)))),
                          naturality := ⋯ } } ≫
                    (pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).map f))
                U) :
            (a : (↑(((pushforward φ).obj X).val.obj U) : Type v)) →
              (↑(((pushforward (φ ≫ (sheafPushforwardContinuousNatTrans α RingCat J K).app S)).obj Y).val.obj U) :
                Type v))
          x :
        (↑(((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj Y).val.obj U) :
          Type v))
⊢ (pushforward φ).map f ≫
      {
        val :=
          {
            app := fun (U : Cᵒᵖ) ↦
              (ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app U))).map
                (Y.val.map (Quiver.Hom.op (α.app (Opposite.unop U)))),
            naturality := ⋯ } } =
    {
        val :=
          {
            app := fun (U : Cᵒᵖ) ↦
              (ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app U))).map
                (X.val.map (Quiver.Hom.op (α.app (Opposite.unop U)))),
            naturality := ⋯ :
            ((pushforward φ).obj X).val.Hom
              ((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val } :
        ((pushforward φ).obj X).Hom
          ((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X) } ≫
      (pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).map f

Tactic:
ext U x