awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ Functor.obj (F.sheafPushforwardContinuous RingCat J K) R
inst_3 :
  J.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_4 :
  K.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_5 : (pushforward φ).IsRightAdjoint
inst_6 : F.Final
M : SheafOfModules R
⊢ Function.Bijective (β := (pullback φ).obj (unit S) ⟶ M) fun (x : unit R ⟶ M) ↦ pullbackObjUnitToUnit φ ≫ x
have rw : Function.Bijective (α := unit R ⟶ M) (β := ((pushforward φ).obj M).sections)
  (Function.comp (β := unit S ⟶ (pushforward φ).obj M)
    (⇑((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
    (Function.comp (β := (pullback φ).obj (unit S) ⟶ M)
      (⇑(Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M) :
        (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
      fun (x : unit R ⟶ M) ↦ pullbackObjUnitToUnit φ ≫ x))
Tactic: rw [← ((pullbackPushforwardAdjunction.{u} φ).homEquiv _ _).bijective.of_comp_iff',
    ← (unitHomEquiv _).bijective.of_comp_iff']
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ Functor.obj (F.sheafPushforwardContinuous RingCat J K) R
inst_3 :
  J.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_4 :
  K.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_5 : (pushforward φ).IsRightAdjoint
inst_6 : F.Final
M : SheafOfModules R
⊢ Function.Bijective (α := unit R ⟶ M) (β := ((pushforward φ).obj M).sections)
    (Function.comp (β := unit S ⟶ (pushforward φ).obj M)
      (⇑((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
      (Function.comp (β := (pullback φ).obj (unit S) ⟶ M)
        (⇑(Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M) :
          (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
        fun (x : unit R ⟶ M) ↦ pullbackObjUnitToUnit φ ≫ x))
have h : Eq (α := (unit R ⟶ M) → ((pushforward φ).obj M).sections)
  (Function.comp (β := unit S ⟶ (pushforward φ).obj M)
    (⇑((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
    (Function.comp (β := (pullback φ).obj (unit S) ⟶ M)
      (⇑(Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M) :
        (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
      fun (x : unit R ⟶ M) ↦ pullbackObjUnitToUnit φ ≫ x))
  (pushforwardSections φ ∘ (⇑M.unitHomEquiv : (a : unit R ⟶ M) → M.sections))
Tactic: convert! (bijective_pushforwardSections φ M).comp (unitHomEquiv _).bijective
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ Functor.obj (F.sheafPushforwardContinuous RingCat J K) R
inst_3 :
  J.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_4 :
  K.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_5 : (pushforward φ).IsRightAdjoint
inst_6 : F.Final
M : SheafOfModules R
⊢ Eq (α := (unit R ⟶ M) → ((pushforward φ).obj M).sections)
    (Function.comp (β := unit S ⟶ (pushforward φ).obj M)
      (⇑((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
      (Function.comp (β := (pullback φ).obj (unit S) ⟶ M)
        (⇑(Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M) :
          (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
        fun (x : unit R ⟶ M) ↦ pullbackObjUnitToUnit φ ≫ x))
    (pushforwardSections φ ∘ (⇑M.unitHomEquiv : (a : unit R ⟶ M) → M.sections))
have h : ∀ (f : unit R ⟶ M),
  Eq (α := ((pushforward φ).obj M).sections)
    (Function.comp (β := unit S ⟶ (pushforward φ).obj M)
      (⇑((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
      (Function.comp (β := (pullback φ).obj (unit S) ⟶ M)
        (⇑(Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M) :
          (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
        fun (x : unit R ⟶ M) ↦ pullbackObjUnitToUnit φ ≫ x)
      f)
    ((pushforwardSections φ ∘ (⇑M.unitHomEquiv : (a : unit R ⟶ M) → M.sections)) f)
Tactic: ext f : 1
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ Functor.obj (F.sheafPushforwardContinuous RingCat J K) R
inst_3 :
  J.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_4 :
  K.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_5 : (pushforward φ).IsRightAdjoint
inst_6 : F.Final
M : SheafOfModules R
f : unit R ⟶ M
⊢ Eq (α := ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
            (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
          (pullbackObjUnitToUnit φ ≫ f)) :
      ((pushforward φ).obj M).sections)
    (pushforwardSections φ ((M.unitHomEquiv : (a : unit R ⟶ M) → M.sections) f) : ((pushforward φ).obj M).sections)
have h : Eq (α := ((pushforward φ).obj M).sections)
  ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
      ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
          (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
        (pullbackObjUnitToUnit φ ≫ f)) :
    ((pushforward φ).obj M).sections)
  ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
      (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f) :
    ((pushforward φ).obj M).sections)
Tactic: rw [pushforwardSections_unitHomEquiv, EmbeddingLike.apply_eq_iff_eq,
    Adjunction.homEquiv_naturality_right,
    pullbackPushforwardAdjunction_homEquiv_pullbackObjUnitToUnit]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ Functor.obj (F.sheafPushforwardContinuous RingCat J K) R
inst_3 :
  J.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_4 :
  K.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_5 : (pushforward φ).IsRightAdjoint
inst_6 : F.Final
M : SheafOfModules R
f : unit R ⟶ M
h :
  Eq (α := ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
            (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
          (pullbackObjUnitToUnit φ ≫ f)) :
      ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f) :
      ((pushforward φ).obj M).sections)
⊢ Eq (α := ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
            (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
          (pullbackObjUnitToUnit φ ≫ f)) :
      ((pushforward φ).obj M).sections)
    (pushforwardSections φ ((M.unitHomEquiv : (a : unit R ⟶ M) → M.sections) f) : ((pushforward φ).obj M).sections)
have h₁ : Eq (α := unit S ⟶ (pushforward φ).obj M)
  ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
        (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
      (pullbackObjUnitToUnit φ ≫ f) :
    unit S ⟶ (pushforward φ).obj M)
  (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f : unit S ⟶ (pushforward φ).obj M)
Tactic: rw [pushforwardSections_unitHomEquiv, EmbeddingLike.apply_eq_iff_eq,
    Adjunction.homEquiv_naturality_right,
    pullbackPushforwardAdjunction_homEquiv_pullbackObjUnitToUnit]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ Functor.obj (F.sheafPushforwardContinuous RingCat J K) R
inst_3 :
  J.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_4 :
  K.HasSheafCompose
    (forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
      AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
      AddCommGrpCat.carrier))
inst_5 : (pushforward φ).IsRightAdjoint
inst_6 : F.Final
M : SheafOfModules R
f : unit R ⟶ M
h :
  Eq (α := ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
            (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
          (pullbackObjUnitToUnit φ ≫ f)) :
      ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f) :
      ((pushforward φ).obj M).sections)
h₁ :
  Eq (α := unit S ⟶ (pushforward φ).obj M)
    ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
          (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
        (pullbackObjUnitToUnit φ ≫ f) :
      unit S ⟶ (pushforward φ).obj M)
    (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f : unit S ⟶ (pushforward φ).obj M)
⊢ Eq (α := ((pushforward φ).obj M).sections)
    ((((pushforward φ).obj M).unitHomEquiv : (a : unit S ⟶ (pushforward φ).obj M) → ((pushforward φ).obj M).sections)
        ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) M :
            (a : (pullback φ).obj (unit S) ⟶ M) → unit S ⟶ (pushforward φ).obj M)
          (pullbackObjUnitToUnit φ ≫ f)) :
      ((pushforward φ).obj M).sections)
    (pushforwardSections φ ((M.unitHomEquiv : (a : unit R ⟶ M) → M.sections) f) : ((pushforward φ).obj M).sections)
have h₂ : Eq (α := unit S ⟶ (pushforward φ).obj M)
  ((Adjunction.homEquiv (pullbackPushforwardAdjunction φ) (unit S) (unit R) :
          (a : (pullback φ).obj (unit S) ⟶ unit R) → unit S ⟶ (pushforward φ).obj (unit R))
        (pullbackObjUnitToUnit φ) ≫
      (pushforward φ).map f :
    unit S ⟶ (pushforward φ).obj M)
  (unitToPushforwardObjUnit φ ≫ (pushforward φ).map f : unit S ⟶ (pushforward φ).obj M)
Tactic: rw [pushforwardSections_unitHomEquiv, EmbeddingLike.apply_eq_iff_eq,
    Adjunction.homEquiv_naturality_right,
    pullbackPushforwardAdjunction_homEquiv_pullbackObjUnitToUnit]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
D' : Type u₃
inst_2 : Category.{v₃, u₃} D'
D'' : Type u₄
inst_3 : Category.{v₄, u₄} D''
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_4 : F.IsContinuous J K
φ_1 φ ψ : S ⟶ obj (F.sheafPushforwardContinuous RingCat J K) R
e : φ = ψ
X : SheafOfModules.{v, v₂, u₂, u} R
U : Cᵒᵖ
⊢ RingCat.Hom.hom (φ.hom.app U) = RingCat.Hom.hom (ψ.hom.app U)
have subst : RingCat.Hom.hom (φ.hom.app U) = RingCat.Hom.hom (φ.hom.app U)
Tactic: subst e
Cleared hypotheses: ψ, e
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
D' : Type u₃
inst_2 : Category.{v₃, u₃} D'
D'' : Type u₄
inst_3 : Category.{v₄, u₄} D''
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_4 : F.IsContinuous J K
φ_1 φ ψ : S ⟶ obj (F.sheafPushforwardContinuous RingCat J K) R
e : φ = ψ
X Y : SheafOfModules.{v, v₂, u₂, u} R
x : X ⟶ Y
⊢ (pushforward φ).map x ≫
      Iso.hom
        ((fun (X : SheafOfModules.{v, v₂, u₂, u} R) ↦
            ((fullyFaithfulForget S).preimageIso
                (PresheafOfModules.isoMk
                  (fun (U : Cᵒᵖ) ↦
                    (ModuleCat.restrictScalarsCongr ⋯).app
                      ((obj (PresheafOfModules.pushforward₀ F (obj (sheafToPresheaf K RingCat) R)) X.val).obj U))
                  ⋯) :
              (pushforward φ).obj X ≅ (pushforward ψ).obj X))
          Y) =
    Iso.hom
        ((fun (X : SheafOfModules.{v, v₂, u₂, u} R) ↦
            ((fullyFaithfulForget S).preimageIso
                (PresheafOfModules.isoMk
                  (fun (U : Cᵒᵖ) ↦
                    (ModuleCat.restrictScalarsCongr ⋯).app
                      ((obj (PresheafOfModules.pushforward₀ F (obj (sheafToPresheaf K RingCat) R)) X.val).obj U))
                  ⋯) :
              (pushforward φ).obj X ≅ (pushforward ψ).obj X))
          X) ≫
      (pushforward ψ).map x
have subst : (pushforward φ).map x ≫
    Iso.hom
      ((fun (X : SheafOfModules.{v, v₂, u₂, u} R) ↦
          ((fullyFaithfulForget S).preimageIso
              (PresheafOfModules.isoMk
                (fun (U : Cᵒᵖ) ↦
                  (ModuleCat.restrictScalarsCongr sorry).app
                    ((obj (PresheafOfModules.pushforward₀ F (obj (sheafToPresheaf K RingCat) R)) X.val).obj U))
                sorry) :
            (pushforward φ).obj X ≅ (pushforward φ).obj X))
        Y) =
  Iso.hom
      ((fun (X : SheafOfModules.{v, v₂, u₂, u} R) ↦
          ((fullyFaithfulForget S).preimageIso
              (PresheafOfModules.isoMk
                (fun (U : Cᵒᵖ) ↦
                  (ModuleCat.restrictScalarsCongr sorry).app
                    ((obj (PresheafOfModules.pushforward₀ F (obj (sheafToPresheaf K RingCat) R)) X.val).obj U))
                sorry) :
            (pushforward φ).obj X ≅ (pushforward φ).obj X))
        X) ≫
    (pushforward φ).map x
Tactic: subst e
Cleared hypotheses: ψ, e
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
D' : Type u₃
inst_2 : Category.{v₃, u₃} D'
D'' : Type u₄
inst_3 : Category.{v₄, u₄} D''
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_4 : F.IsContinuous J K
φ : S ⟶ obj (F.sheafPushforwardContinuous RingCat J K) R
K' : GrothendieckTopology.{v₃, u₃} D'
K'' : GrothendieckTopology.{v₄, u₄} D''
G : CategoryTheory.Functor.{v₂, v₃, u₂, u₃} D D'
R' : Sheaf K' RingCat
inst_5 : G.IsContinuous K K'
inst_6 : (F ⋙ G).IsContinuous J K'
ψ : R ⟶ obj (G.sheafPushforwardContinuous RingCat K K') R'
G' : CategoryTheory.Functor.{v₃, v₄, u₃, u₄} D' D''
R'' : Sheaf K'' RingCat
inst_7 : G'.IsContinuous K' K''
inst_8 : (G ⋙ G').IsContinuous K K''
inst_9 : ((F ⋙ G) ⋙ G').IsContinuous J K''
inst_10 : (F ⋙ G ⋙ G').IsContinuous J K''
ψ' : R' ⟶ obj (G'.sheafPushforwardContinuous RingCat K' K'') R''
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have w : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ obj (F.sheafPushforwardContinuous RingCat J K) R
⊢ pushforwardComp (𝟙 S) φ = (pushforward φ).isoWhiskerLeft (pushforwardId S) ≪≫ (pushforward φ).rightUnitor
have w : ∀ (x : SheafOfModules.{v, v₂, u₂, u} R) (X : Cᵒᵖ)
  (x_1 : (↑(PresheafOfModules.obj (val (obj (pushforward φ ⋙ pushforward (F := 𝟭 C) (𝟙 S)) x)) X) : Type v)),
  Eq (α :=
    (↑((obj
                (pushforward (F := 𝟭 C ⋙ F)
                  (𝟙 S ≫ CategoryTheory.Functor.map (sheafPushforwardContinuous (𝟭 C) RingCat J J) φ))
                x).val.obj
          X) :
      Type v))
    ((ModuleCat.Hom.hom (PresheafOfModules.Hom.app (Hom.val (NatTrans.app (Iso.hom (pushforwardComp (𝟙 S) φ)) x)) X) :
          (a : (↑(((pushforward φ ⋙ pushforward (𝟙 S)).obj x).val.obj X) : Type v)) →
            (↑(((pushforward (𝟙 S ≫ ((𝟭 C).sheafPushforwardContinuous RingCat J J).map φ)).obj x).val.obj X) : Type v))
        x_1 :
      (↑((obj
                  (pushforward (F := 𝟭 C ⋙ F)
                    (𝟙 S ≫ CategoryTheory.Functor.map (sheafPushforwardContinuous (𝟭 C) RingCat J J) φ))
                  x).val.obj
            X) :
        Type v))
    ((ModuleCat.Hom.hom
            ((((pushforward φ).isoWhiskerLeft (pushforwardId S) ≪≫ (pushforward φ).rightUnitor).hom.app x).val.app X) :
          (a : (↑(((pushforward φ ⋙ pushforward (𝟙 S)).obj x).val.obj X) : Type v)) →
            (↑(((pushforward (𝟙 S ≫ ((𝟭 C).sheafPushforwardContinuous RingCat J J).map φ)).obj x).val.obj X) : Type v))
        x_1 :
      (↑((obj
                  (pushforward (F := 𝟭 C ⋙ F)
                    (𝟙 S ≫ CategoryTheory.Functor.map (sheafPushforwardContinuous (𝟭 C) RingCat J J) φ))
                  x).val.obj
            X) :
        Type v))
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst_2 : F.IsContinuous J K
φ : S ⟶ obj (F.sheafPushforwardContinuous RingCat J K) R
⊢ pushforwardComp φ (𝟙 R) = isoWhiskerRight (pushforwardId R) (pushforward φ) ≪≫ (pushforward φ).leftUnitor
have w : ∀ (x : SheafOfModules.{v, v₂, u₂, u} R) (X : Cᵒᵖ)
  (x_1 : (↑((obj (pushforward (F := 𝟭 D) (𝟙 R) ⋙ pushforward φ) x).val.obj X) : Type v)),
  Eq (α :=
    (↑(((pushforward (φ ≫ CategoryTheory.Functor.map (F.sheafPushforwardContinuous RingCat J K) (𝟙 R))).obj x).val.obj
          X) :
      Type v))
    ((ModuleCat.Hom.hom (PresheafOfModules.Hom.app (Hom.val (NatTrans.app (Iso.hom (pushforwardComp φ (𝟙 R))) x)) X) :
          (a : (↑(((pushforward (𝟙 R) ⋙ pushforward φ).obj x).val.obj X) : Type v)) →
            (↑(((pushforward (φ ≫ (F.sheafPushforwardContinuous RingCat J K).map (𝟙 R))).obj x).val.obj X) : Type v))
        x_1 :
      (↑(((pushforward (φ ≫ CategoryTheory.Functor.map (F.sheafPushforwardContinuous RingCat J K) (𝟙 R))).obj x).val.obj
            X) :
        Type v))
    ((ModuleCat.Hom.hom
            (((isoWhiskerRight (pushforwardId R) (pushforward φ) ≪≫ (pushforward φ).leftUnitor).hom.app x).val.app X) :
          (a : (↑(((pushforward (𝟙 R) ⋙ pushforward φ).obj x).val.obj X) : Type v)) →
            (↑(((pushforward (φ ≫ (F.sheafPushforwardContinuous RingCat J K).map (𝟙 R))).obj x).val.obj X) : Type v))
        x_1 :
      (↑(((pushforward (φ ≫ CategoryTheory.Functor.map (F.sheafPushforwardContinuous RingCat J K) (𝟙 R))).obj x).val.obj
            X) :
        Type v))
Tactic: ext