-- 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
h :
  ∀ (U : Dᵒᵖ) (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))
⊢ ((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) =
    𝟙 ((pushforward ψ).obj X)

Tactic:
ext U x