-- 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
x : (↑(((pushforward φ).obj X).val.obj U) : Type v)
hf :
  Eq (α := (↑(X.val.presheaf.obj (Opposite.op (F.obj (Opposite.unop V)))) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier)
            (X.val.presheaf.map (G.map i.unop).op ≫ X.val.presheaf.map (Quiver.Hom.op (α.app (Opposite.unop V)))) :
          (a : (↑(X.val.presheaf.obj (Opposite.op (G.obj (Opposite.unop U)))) : Type v)) →
            (↑(X.val.presheaf.obj (Opposite.op (F.obj (Opposite.unop V)))) : Type v))
        x :
      (↑(X.val.presheaf.obj (Opposite.op (F.obj (Opposite.unop V)))) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier)
            (X.val.presheaf.map (Quiver.Hom.op (α.app (Opposite.unop U))) ≫ X.val.presheaf.map (F.map i.unop).op) :
          (a : (↑(X.val.presheaf.obj (Opposite.op (G.obj (Opposite.unop U)))) : Type v)) →
            (↑(X.val.presheaf.obj (Opposite.op (F.obj (Opposite.unop V)))) : Type v))
        x :
      (↑(X.val.presheaf.obj (Opposite.op (F.obj (Opposite.unop V)))) : 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))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat.{v, u} (↑(T.obj (P := Presheaf.IsSheaf J).obj V) : Type u)) ↦
            LinearMap.{u, u, v, v} (R := (↑(T.obj (P := Presheaf.IsSheaf J).obj V) : Type u))
              (RingHom.id (↑(T.obj (P := Presheaf.IsSheaf J).obj V) : Type u)) (↑x1 : Type v) (↑x2 : Type v))
            (CC := ModuleCat.carrier) (Y :=
            ((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val.obj V)
            ((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 V) : Type v)) →
            (↑(((pushforward (φ ≫ (sheafPushforwardContinuousNatTrans α RingCat J K).app S)).obj X).val.obj V) :
              Type v))
        ((ModuleCat.Hom.hom (((pushforward φ).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).obj X).val.obj V)) :
                  Type v))
            x :
          (↑((ModuleCat.restrictScalars (RingCat.Hom.hom (T.obj (P := Presheaf.IsSheaf J).map i))).obj
                (((pushforward φ).obj X).val.obj V)) :
            Type v)) :
      (↑(((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val.obj V) :
        Type v))
    ((ModuleCat.Hom.hom
            (((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val.map i) :
          (a :
              (↑(((pushforward (φ ≫ (sheafPushforwardContinuousNatTrans α RingCat J K).app S)).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))
        ((ModuleCat.Hom.hom
                ((ModuleCat.restrictScalars (RingCat.Hom.hom (φ.hom.app U))).map
                  (X.val.map (Quiver.Hom.op (α.app (Opposite.unop U))))) :
              (a : (↑(((pushforward φ).obj X).val.obj U) : Type v)) →
                (↑(((pushforward (φ ≫ (sheafPushforwardContinuousNatTrans α RingCat J K).app S)).obj X).val.obj U) :
                  Type v))
            x :
          (↑(((pushforward (φ ≫ NatTrans.app (sheafPushforwardContinuousNatTrans α RingCat J K) S)).obj X).val.obj U) :
            Type v)) :
      (↑((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))

Tactic:
change (X.val.presheaf.map (G.map i.unop).op ≫ X.val.presheaf.map (α.app V.unop).op) _ =
        (X.val.presheaf.map (α.app U.unop).op ≫ X.val.presheaf.map (F.map i.unop).op) _

Premises used: CategoryTheory.ConcreteCategory.hom, Quiver.Hom.op, Quiver.Hom.unop, SheafOfModules.val, CategoryTheory.Functor.map, CategoryTheory.CategoryStruct.comp, DFunLike.coe, Opposite.unop, CategoryTheory.NatTrans.app, PresheafOfModules.presheaf