awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Free (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
F : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ (Type u)
G : PresheafOfModules R
ψ : freeObj F ⟶ G
X : Cᵒᵖ
⊢ ModuleCat.freeDesc
      (CategoryStruct.comp (Y := (↑(Functor.obj (ModuleCat.free (↑(R.obj X) : Type u)) (F.obj X)) : Type u))
        (↾fun (x : F.obj X) ↦ ModuleCat.freeMk x)
        (↾(⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
                AddCommGrpCat.carrier) (((toPresheaf R).map ψ).app X)) :
            (a : (↑((ModuleCat.free (↑(R.obj X) : Type u)).obj (F.obj X)) : Type u)) → (↑(G.obj X) : Type u)))) =
    ψ.app X
have h : ∀ (x : F.obj X),
  Eq (α := (↑(G.obj X) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑(R.obj X) : Type u)) ↦
            LinearMap (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type u) (↑x2 : Type u))
            (CC := ModuleCat.carrier) (X := Functor.obj (ModuleCat.free (↑(R.obj X) : Type u)) (F.obj X)) (Y := G.obj X)
            (ModuleCat.freeDesc
              (CategoryStruct.comp (Y := (↑(Functor.obj (ModuleCat.free (↑(R.obj X) : Type u)) (F.obj X)) : Type u))
                (↾fun (x : F.obj X) ↦ ModuleCat.freeMk x)
                (↾(⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u))
                        (CC := AddCommGrpCat.carrier) (((toPresheaf R).map ψ).app X)) :
                    (a : (↑((ModuleCat.free (↑(R.obj X) : Type u)).obj (F.obj X)) : Type u)) →
                      (↑(G.obj X) : Type u))))) :
          (a : (↑((ModuleCat.free (↑(R.obj X) : Type u)).obj (F.obj X)) : Type u)) → (↑(G.obj X) : Type u))
        (ModuleCat.freeMk x) :
      (↑(G.obj X) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑(R.obj X) : Type u)) ↦
            LinearMap (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type u) (↑x2 : Type u))
            (CC := ModuleCat.carrier) (ψ.app X) :
          (a : (↑((ModuleCat.free (↑(R.obj X) : Type u)).obj (F.obj X)) : Type u)) → (↑(G.obj X) : Type u))
        (ModuleCat.freeMk x) :
      (↑(G.obj X) : Type u))
Tactic: ext x
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Free (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
F : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ (Type u)
G : PresheafOfModules R
φ :
  F ⟶
    G.presheaf ⋙
      forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC := AddCommGrpCat.carrier)
⊢ (fun (ψ : freeObj F ⟶ G) ↦
        freeAdjunctionUnit R F ≫
          Functor.whiskerRight ((toPresheaf R).map ψ)
            (forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
              AddCommGrpCat.carrier)))
      ((fun
            (φ :
              F ⟶
                G.presheaf ⋙
                  forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
                    AddCommGrpCat.carrier)) ↦
          freeObjDesc φ)
        φ) =
    φ
have w : ∀ (x : Cᵒᵖ) (x_1 : F.obj x),
  TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          ((fun (ψ : freeObj F ⟶ G) ↦
              freeAdjunctionUnit R F ≫
                Functor.whiskerRight ((toPresheaf R).map ψ)
                  (forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
                    AddCommGrpCat.carrier)))
            ((fun
                  (φ :
                    F ⟶
                      G.presheaf ⋙
                        forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
                          AddCommGrpCat.carrier)) ↦
                freeObjDesc φ)
              φ))
          x))
      x_1 =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (φ.app x)).toFun x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Free (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
F : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ (Type u)
G_1 : PresheafOfModules R
F₁ F₂ : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ (Type u)
G : PresheafOfModules R
f : F₁ ⟶ F₂
g :
  F₂ ⟶
    (toPresheaf R ⋙
          Functor.obj (Functor.whiskeringRight Cᵒᵖ Ab (Type u))
            (forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
              AddCommGrpCat.carrier))).obj
      G
⊢ Eq (α :=
    F₁ ⟶
      (toPresheaf R).obj G ⋙
        forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
          AddCommGrpCat.carrier))
    (freeAdjunctionUnit R F₁ ≫
      Functor.whiskerRight
        ((toPresheaf R).map
          ((Equiv.symm freeHomEquiv :
              (a : F₁ ⟶ (toPresheaf R ⋙ (Functor.whiskeringRight Cᵒᵖ Ab (Type u)).obj (forget Ab)).obj G) →
                (free R).obj F₁ ⟶ G)
            (f ≫ g)))
        (forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
          AddCommGrpCat.carrier)))
    (freeAdjunctionUnit R F₁ ≫
      Functor.whiskerRight
        ((toPresheaf R).map
          ((free R).map f ≫
            (Equiv.symm freeHomEquiv :
                (a : F₂ ⟶ (toPresheaf R ⋙ (Functor.whiskeringRight Cᵒᵖ Ab (Type u)).obj (forget Ab)).obj G) →
                  (free R).obj F₂ ⟶ G)
              g))
        (forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
          AddCommGrpCat.carrier)))
have w : ∀ (x : Cᵒᵖ) (x_1 : F₁.obj x),
  Eq (α :=
    ((toPresheaf R).obj G ⋙
          forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
            AddCommGrpCat.carrier)).obj
      x)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (freeAdjunctionUnit R F₁ ≫
            Functor.whiskerRight
              ((toPresheaf R).map
                ((Equiv.symm freeHomEquiv :
                    (a : F₁ ⟶ (toPresheaf R ⋙ (Functor.whiskeringRight Cᵒᵖ Ab (Type u)).obj (forget Ab)).obj G) →
                      (free R).obj F₁ ⟶ G)
                  (f ≫ g)))
              (forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
                AddCommGrpCat.carrier)))
          x))
      x_1)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (freeAdjunctionUnit R F₁ ≫
            Functor.whiskerRight
              ((toPresheaf R).map
                ((free R).map f ≫
                  (Equiv.symm freeHomEquiv :
                      (a : F₂ ⟶ (toPresheaf R ⋙ (Functor.whiskeringRight Cᵒᵖ Ab (Type u)).obj (forget Ab)).obj G) →
                        (free R).obj F₂ ⟶ G)
                    g))
              (forget Ab (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
                AddCommGrpCat.carrier)))
          x))
      x_1)
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Generator (goal)
Goal:
C : Type u
inst : Category.{v, u} C
R : CategoryTheory.Functor.{v, v, u, v + 1} Cᵒᵖ RingCat
M N : PresheafOfModules R
f₁ f₂ : M ⟶ N
h : ∀ (G : PresheafOfModules R), freeYoneda R G → ∀ (h : G ⟶ M), h ≫ f₁ = h ≫ f₂
⊢ f₁ = f₂
have h₁ : ∀ (X : C) (m : (↑(M.obj (Opposite.op X)) : Type v)),
  Eq (α := (↑(N.obj (Opposite.op X)) : Type v))
    ((ModuleCat.Hom.hom (f₁.app (Opposite.op X)) :
          (a : (↑(M.obj (Opposite.op X)) : Type v)) → (↑(N.obj (Opposite.op X)) : Type v))
        m :
      (↑(N.obj (Opposite.op X)) : Type v))
    ((ModuleCat.Hom.hom (f₂.app (Opposite.op X)) :
          (a : (↑(M.obj (Opposite.op X)) : Type v)) → (↑(N.obj (Opposite.op X)) : Type v))
        m :
      (↑(N.obj (Opposite.op X)) : Type v))
Tactic: ext ⟨X⟩ m
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Generator (goal)
Goal:
C : Type u
inst : Category.{v, u} C
R : CategoryTheory.Functor.{v, v, u, v + 1} Cᵒᵖ RingCat
M N : PresheafOfModules R
f₁ f₂ : M ⟶ N
h : ∀ (G : PresheafOfModules R), freeYoneda R G → ∀ (h : G ⟶ M), h ≫ f₁ = h ≫ f₂
X : C
m : (↑(M.obj (Opposite.op X)) : Type v)
⊢ Eq (α := (↑(N.obj (Opposite.op X)) : Type v))
    ((ModuleCat.Hom.hom (f₁.app (Opposite.op X)) :
          (a : (↑(M.obj (Opposite.op X)) : Type v)) → (↑(N.obj (Opposite.op X)) : Type v))
        m :
      (↑(N.obj (Opposite.op X)) : Type v))
    ((ModuleCat.Hom.hom (f₂.app (Opposite.op X)) :
          (a : (↑(M.obj (Opposite.op X)) : Type v)) → (↑(N.obj (Opposite.op X)) : Type v))
        m :
      (↑(N.obj (Opposite.op X)) : Type v))
have h₁ : ∀ (g : (free R).obj (Functor.obj yoneda X) ⟶ M),
  Eq (α := (↑(N.obj (Opposite.op X)) : Type v))
    ((ModuleCat.Hom.hom (f₁.app (Opposite.op X)) :
          (a : (↑(M.obj (Opposite.op X)) : Type v)) → (↑(N.obj (Opposite.op X)) : Type v))
        ((freeYonedaEquiv : (a : (free R).obj (yoneda.obj X) ⟶ M) → (↑(M.obj (Opposite.op X)) : Type v)) g :
          (↑(M.obj (Opposite.op X)) : Type v)) :
      (↑(N.obj (Opposite.op X)) : Type v))
    ((ModuleCat.Hom.hom (f₂.app (Opposite.op X)) :
          (a : (↑(M.obj (Opposite.op X)) : Type v)) → (↑(N.obj (Opposite.op X)) : Type v))
        ((freeYonedaEquiv : (a : (free R).obj (yoneda.obj X) ⟶ M) → (↑(M.obj (Opposite.op X)) : Type v)) g :
          (↑(M.obj (Opposite.op X)) : Type v)) :
      (↑(N.obj (Opposite.op X)) : Type v))
Tactic: obtain ⟨g, rfl⟩ := freeYonedaEquiv.surjective m
Cleared hypotheses: m
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
J : Type u₂
inst_1 : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules.{v, v₁, u₁, u} R
inst_2 :
  ∀ (X : Cᵒᵖ),
    Small.{v, max u₂ v}
      (↑((F ⋙ evaluation R X) ⋙
              forget (ModuleCat.{v, u} (↑(R.obj X) : Type u)) (FC :=
                fun (x1 x2 : ModuleCat.{v, u} (↑(R.obj X) : Type u)) ↦
                LinearMap.{u, u, v, v} (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type v)
                  (↑x2 : Type v))
                (CC := ModuleCat.carrier)).sections :
        Type (max u₂ v))
c : Cone F
hc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone c)
s : Cone F
X Y : Cᵒᵖ
f : X ⟶ Y
⊢ Eq (α := s.pt.obj X ⟶ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).obj (c.pt.obj Y))
    (s.pt.map f ≫
      (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((hc Y).lift ((evaluation R Y).mapCone s)))
    ((hc X).lift ((evaluation R X).mapCone s) ≫ c.pt.map f)
have apply : ∀ (j : J),
  (s.pt.map f ≫
        (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((hc Y).lift ((evaluation R Y).mapCone s))) ≫
      ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).mapCone ((evaluation R Y).mapCone c)).π.app j =
    ((hc X).lift ((evaluation R X).mapCone s) ≫ c.pt.map f) ≫
      ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).mapCone ((evaluation R Y).mapCone c)).π.app j
Tactic: apply (isLimitOfPreserves (ModuleCat.restrictScalars (R.map f).hom) (hc Y)).hom_ext
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
J : Type u₂
inst_1 : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules.{v, v₁, u₁, u} R
inst_2 :
  ∀ (X : Cᵒᵖ),
    Small.{v, max u₂ v}
      (↑((F ⋙ evaluation R X) ⋙
              forget (ModuleCat.{v, u} (↑(R.obj X) : Type u)) (FC :=
                fun (x1 x2 : ModuleCat.{v, u} (↑(R.obj X) : Type u)) ↦
                LinearMap.{u, u, v, v} (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type v)
                  (↑x2 : Type v))
                (CC := ModuleCat.carrier)).sections :
        Type (max u₂ v))
c : Cone F
hc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone c)
s : Cone F
X Y : Cᵒᵖ
f : X ⟶ Y
j : J
⊢ (s.pt.map f ≫
        (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((hc Y).lift ((evaluation R Y).mapCone s))) ≫
      ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).mapCone ((evaluation R Y).mapCone c)).π.app j =
    ((hc X).lift ((evaluation R X).mapCone s) ≫ c.pt.map f) ≫
      ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).mapCone ((evaluation R Y).mapCone c)).π.app j
have h₁ : (Functor.obj (Functor.obj (Functor.const J) c.pt) j).map f ≫
    (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y) =
  (c.π.app j).app X ≫ (F.obj j).map f
Tactic: have h₁ := (c.π.app j).naturality f
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
J : Type u₂
inst_1 : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules.{v, v₁, u₁, u} R
inst_2 :
  ∀ (X : Cᵒᵖ),
    Small.{v, max u₂ v}
      (↑((F ⋙ evaluation R X) ⋙
              forget (ModuleCat.{v, u} (↑(R.obj X) : Type u)) (FC :=
                fun (x1 x2 : ModuleCat.{v, u} (↑(R.obj X) : Type u)) ↦
                LinearMap.{u, u, v, v} (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type v)
                  (↑x2 : Type v))
                (CC := ModuleCat.carrier)).sections :
        Type (max u₂ v))
c : Cone F
hc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone c)
s : Cone F
X Y : Cᵒᵖ
f : X ⟶ Y
j : J
h₁ :
  (Functor.obj (Functor.obj (Functor.const J) c.pt) j).map f ≫
      (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y) =
    (c.π.app j).app X ≫ (F.obj j).map f
⊢ (s.pt.map f ≫
        (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((hc Y).lift ((evaluation R Y).mapCone s))) ≫
      ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).mapCone ((evaluation R Y).mapCone c)).π.app j =
    ((hc X).lift ((evaluation R X).mapCone s) ≫ c.pt.map f) ≫
      ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).mapCone ((evaluation R Y).mapCone c)).π.app j
have h₂ : (hc X).lift ((evaluation R X).mapCone s) ≫ ((evaluation R X).mapCone c).π.app j = ((evaluation R X).mapCone s).π.app j
Tactic: have h₂ := (hc X).fac ((evaluation R X).mapCone s) j
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
J : Type u₂
inst_1 : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules.{v, v₁, u₁, u} R
inst_2 :
  ∀ (X : Cᵒᵖ),
    Small.{v, max u₂ v}
      (↑((F ⋙ evaluation R X) ⋙
              forget (ModuleCat.{v, u} (↑(R.obj X) : Type u)) (FC :=
                fun (x1 x2 : ModuleCat.{v, u} (↑(R.obj X) : Type u)) ↦
                LinearMap.{u, u, v, v} (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type v)
                  (↑x2 : Type v))
                (CC := ModuleCat.carrier)).sections :
        Type (max u₂ v))
c : Cone F
hc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone c)
s : Cone F
X Y : Cᵒᵖ
f : X ⟶ Y
j : J
h₁ :
  (Functor.obj (Functor.obj (Functor.const J) c.pt) j).map f ≫
      (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y) =
    (c.π.app j).app X ≫ (F.obj j).map f
h₂ :
  (hc X).lift ((evaluation R X).mapCone s) ≫ ((evaluation R X).mapCone c).π.app j = ((evaluation R X).mapCone s).π.app j
⊢ (s.pt.map f ≫
        (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((hc Y).lift ((evaluation R Y).mapCone s))) ≫
      ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).mapCone ((evaluation R Y).mapCone c)).π.app j =
    ((hc X).lift ((evaluation R X).mapCone s) ≫ c.pt.map f) ≫
      ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).mapCone ((evaluation R Y).mapCone c)).π.app j
have rw : s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map (((evaluation R Y).mapCone s).π.app j) =
  (hc X).lift ((evaluation R X).mapCone s) ≫
    c.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map (((evaluation R Y).mapCone c).π.app j)
Tactic: rw [Functor.mapCone_π_app, assoc, assoc, ← Functor.map_comp, IsLimit.fac]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
J : Type u₂
inst_1 : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules.{v, v₁, u₁, u} R
inst_2 :
  ∀ (X : Cᵒᵖ),
    Small.{v, max u₂ v}
      (↑((F ⋙ evaluation R X) ⋙
              forget (ModuleCat.{v, u} (↑(R.obj X) : Type u)) (FC :=
                fun (x1 x2 : ModuleCat.{v, u} (↑(R.obj X) : Type u)) ↦
                LinearMap.{u, u, v, v} (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type v)
                  (↑x2 : Type v))
                (CC := ModuleCat.carrier)).sections :
        Type (max u₂ v))
c : Cone F
hc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone c)
s : Cone F
X Y : Cᵒᵖ
f : X ⟶ Y
j : J
h₁ :
  (Functor.obj (Functor.obj (Functor.const J) c.pt) j).map f ≫
      (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y) =
    (c.π.app j).app X ≫ (F.obj j).map f
h₂ :
  (hc X).lift ((evaluation R X).mapCone s) ≫ ((evaluation R X).mapCone c).π.app j = ((evaluation R X).mapCone s).π.app j
⊢ s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map (((evaluation R Y).mapCone s).π.app j) =
    (hc X).lift ((evaluation R X).mapCone s) ≫
      c.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map (((evaluation R Y).mapCone c).π.app j)
have dsimp : s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((s.π.app j).app Y) =
  (hc X).lift ((evaluation R X).mapCone s) ≫
    c.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y)
Tactic: dsimp at h₁ h₂ ⊢
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
J : Type u₂
inst_1 : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules.{v, v₁, u₁, u} R
inst_2 :
  ∀ (X : Cᵒᵖ),
    Small.{v, max u₂ v}
      (↑((F ⋙ evaluation R X) ⋙
              forget (ModuleCat.{v, u} (↑(R.obj X) : Type u)) (FC :=
                fun (x1 x2 : ModuleCat.{v, u} (↑(R.obj X) : Type u)) ↦
                LinearMap.{u, u, v, v} (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type v)
                  (↑x2 : Type v))
                (CC := ModuleCat.carrier)).sections :
        Type (max u₂ v))
c : Cone F
hc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone c)
s : Cone F
X Y : Cᵒᵖ
f : X ⟶ Y
j : J
h₁ :
  c.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y) =
    (c.π.app j).app X ≫ (F.obj j).map f
h₂ : (hc X).lift ((evaluation R X).mapCone s) ≫ (c.π.app j).app X = (s.π.app j).app X
⊢ s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((s.π.app j).app Y) =
    (hc X).lift ((evaluation R X).mapCone s) ≫
      c.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y)
have rw : s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((s.π.app j).app Y) =
  (hc X).lift ((evaluation R X).mapCone s) ≫ (c.π.app j).app X ≫ (F.obj j).map f
Tactic: rw [h₁, reassoc_of% h₂, Hom.naturality]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
J : Type u₂
inst_1 : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules.{v, v₁, u₁, u} R
inst_2 :
  ∀ (X : Cᵒᵖ),
    Small.{v, max u₂ v}
      (↑((F ⋙ evaluation R X) ⋙
              forget (ModuleCat.{v, u} (↑(R.obj X) : Type u)) (FC :=
                fun (x1 x2 : ModuleCat.{v, u} (↑(R.obj X) : Type u)) ↦
                LinearMap.{u, u, v, v} (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type v)
                  (↑x2 : Type v))
                (CC := ModuleCat.carrier)).sections :
        Type (max u₂ v))
c : Cone F
hc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone c)
s : Cone F
X Y : Cᵒᵖ
f : X ⟶ Y
j : J
h₁ :
  c.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y) =
    (c.π.app j).app X ≫ (F.obj j).map f
h₂ : (hc X).lift ((evaluation R X).mapCone s) ≫ (c.π.app j).app X = (s.π.app j).app X
rw :
  s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((s.π.app j).app Y) =
    (hc X).lift ((evaluation R X).mapCone s) ≫ (c.π.app j).app X ≫ (F.obj j).map f
⊢ s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((s.π.app j).app Y) =
    (hc X).lift ((evaluation R X).mapCone s) ≫
      c.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y)
have rw₁ : s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((s.π.app j).app Y) =
  (s.π.app j).app X ≫ (F.obj j).map f
Tactic: rw [h₁, reassoc_of% h₂, Hom.naturality]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
J : Type u₂
inst_1 : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules.{v, v₁, u₁, u} R
inst_2 :
  ∀ (X : Cᵒᵖ),
    Small.{v, max u₂ v}
      (↑((F ⋙ evaluation R X) ⋙
              forget (ModuleCat.{v, u} (↑(R.obj X) : Type u)) (FC :=
                fun (x1 x2 : ModuleCat.{v, u} (↑(R.obj X) : Type u)) ↦
                LinearMap.{u, u, v, v} (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type v)
                  (↑x2 : Type v))
                (CC := ModuleCat.carrier)).sections :
        Type (max u₂ v))
c : Cone F
hc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone c)
s : Cone F
X Y : Cᵒᵖ
f : X ⟶ Y
j : J
h₁ :
  c.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y) =
    (c.π.app j).app X ≫ (F.obj j).map f
h₂ : (hc X).lift ((evaluation R X).mapCone s) ≫ (c.π.app j).app X = (s.π.app j).app X
rw :
  s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((s.π.app j).app Y) =
    (hc X).lift ((evaluation R X).mapCone s) ≫ (c.π.app j).app X ≫ (F.obj j).map f
rw₁ :
  s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((s.π.app j).app Y) =
    (s.π.app j).app X ≫ (F.obj j).map f
⊢ s.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((s.π.app j).app Y) =
    (hc X).lift ((evaluation R X).mapCone s) ≫
      c.pt.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map ((c.π.app j).app Y)
have rw₂ : (s.π.app j).app X ≫ (F.obj j).map f = (s.π.app j).app X ≫ (F.obj j).map f
Tactic: rw [h₁, reassoc_of% h₂, Hom.naturality]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
R : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
J : Type u₂
inst_1 : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules.{v, v₁, u₁, u} R
inst_2 :
  ∀ (X : Cᵒᵖ),
    Small.{v, max u₂ v}
      (↑((F ⋙ evaluation R X) ⋙
              forget (ModuleCat.{v, u} (↑(R.obj X) : Type u)) (FC :=
                fun (x1 x2 : ModuleCat.{v, u} (↑(R.obj X) : Type u)) ↦
                LinearMap.{u, u, v, v} (R := (↑(R.obj X) : Type u)) (RingHom.id (↑(R.obj X) : Type u)) (↑x1 : Type v)
                  (↑x2 : Type v))
                (CC := ModuleCat.carrier)).sections :
        Type (max u₂ v))
c : Cone F
hc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone c)
s : Cone F
j : J
⊢ { app := fun (X : Cᵒᵖ) ↦ (hc X).lift ((evaluation R X).mapCone s), naturality := ⋯ } ≫ c.π.app j = s.π.app j
have h : ∀ (X : Cᵒᵖ),
  Hom.app ({ app := fun (X : Cᵒᵖ) ↦ (hc X).lift ((evaluation R X).mapCone s), naturality := sorry } ≫ c.π.app j) X =
    (s.π.app j).app X
Tactic: ext1 X