-- 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_1 : 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_1
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 : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
G : CategoryTheory.Functor.{v₂, v₁, u₂, u₁} D C
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_7 : F.IsContinuous J K
inst_8 : G.IsContinuous K J
adj : F ⊣ G
φ : S ⟶ obj (F.sheafPushforwardContinuous RingCat J K) R
ψ : R ⟶ obj (G.sheafPushforwardContinuous RingCat K J) S
H₁ : whiskerRight (NatTrans.op adj.counit) (R.obj (P := Presheaf.IsSheaf K)) = ψ.hom ≫ G.op.whiskerLeft φ.hom
H₂ :
  φ.hom ≫ F.op.whiskerLeft ψ.hom ≫ whiskerRight (NatTrans.op adj.unit) (S.obj (P := Presheaf.IsSheaf J)) =
    𝟙 (S.obj (P := Presheaf.IsSheaf J))
X : SheafOfModules.{v, v₁, u₁, u} S
U : Dᵒᵖ
x : (↑(((pushforward ψ).obj X).val.obj U) : Type v)
h :
  Eq (α := (↑(X.val.presheaf.obj (Opposite.op (obj.{v₁, v₁, u₁, u₁} (𝟭 C) (G.obj (Opposite.unop U))))) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X :=
            X.val.presheaf.obj (Opposite.op (G.obj (obj.{v₂, v₂, u₂, u₂} (𝟭 D) (Opposite.unop U)))))
            (X.val.presheaf.map (Quiver.Hom.op (G.map (adj.counit.app (Opposite.unop U)))) ≫
              X.val.presheaf.map (Quiver.Hom.op (adj.unit.app (G.obj (Opposite.unop U))))) :
          (a : (↑(X.val.presheaf.obj (Opposite.op (G.obj ((𝟭 D).obj (Opposite.unop U))))) : Type v)) →
            (↑(X.val.presheaf.obj (Opposite.op ((𝟭 C).obj (G.obj (Opposite.unop U))))) : Type v))
        x :
      (↑(X.val.presheaf.obj (Opposite.op (obj.{v₁, v₁, u₁, u₁} (𝟭 C) (G.obj (Opposite.unop U))))) : Type v))
    ((ModuleCat.Hom.hom ((Hom.val (𝟙 ((pushforward ψ).obj X))).app U) :
          (a : (↑(((pushforward ψ).obj X).val.obj U) : Type v)) → (↑(((pushforward ψ).obj X).val.obj U) : Type v))
        x :
      (↑(((pushforward ψ).obj X).val.obj U) : Type v))
⊢ Eq (α := (↑(((pushforward ψ).obj X).val.obj U) : Type v))
    (((ModuleCat.Hom.hom
              ((((pushforwardId R).inv ≫
                            pushforwardNatTrans (𝟙 (obj (sheafPushforwardContinuous (𝟭 D) RingCat K K) R)) adj.counit ≫
                              (pushforwardCongr ⋯).hom ≫ Iso.inv (pushforwardComp ψ φ)).app
                        ((pushforward ψ).obj X) ≫
                      (pushforward ψ).map
                        ((Iso.hom (pushforwardComp φ ψ) ≫
                              pushforwardNatTrans
                                  (φ ≫ CategoryTheory.Functor.map (F.sheafPushforwardContinuous RingCat J K) ψ)
                                  adj.unit ≫
                                (pushforwardCongr ⋯).hom ≫ (pushforwardId S).hom).app
                          X)).val.app
                U) :
            LinearMap.{u, u, v, v} (R := (↑(R.obj (P := Presheaf.IsSheaf K).obj U) : Type u))
              (RingHom.id (↑(R.obj (P := Presheaf.IsSheaf K).obj U) : Type u))
              (↑(((pushforward ψ).obj X).val.obj U) : Type v) (↑(((pushforward ψ).obj X).val.obj U) : Type v)) :
          (a : (↑(((pushforward ψ).obj X).val.obj U) : Type v)) → (↑(((pushforward ψ).obj X).val.obj U) : Type v))
        x :
      (↑(((pushforward ψ).obj X).val.obj U) : Type v))
    ((ModuleCat.Hom.hom ((Hom.val (𝟙 ((pushforward ψ).obj X))).app U) :
          (a : (↑(((pushforward ψ).obj X).val.obj U) : Type v)) → (↑(((pushforward ψ).obj X).val.obj U) : Type v))
        x :
      (↑(((pushforward ψ).obj X).val.obj U) : Type v))

Tactic:
change (X.val.presheaf.map (G.map (adj.counit.app U.unop)).op ≫
      X.val.presheaf.map (adj.unit.app (G.obj U.unop)).op) _ = _

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