awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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
X : Cᵒᵖ
⊢ Hom.app ({ app := fun (X : Cᵒᵖ) ↦ (hc X).lift ((evaluation R X).mapCone s), naturality := ⋯ } ≫ c.π.app j) X =
    (s.π.app j).app X

Tactic:
exact (hc X).fac ((evaluation R X).mapCone s) j

Premises used: PresheafOfModules.evaluation, CategoryTheory.Functor.mapCone, CategoryTheory.Limits.IsLimit.fac
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
h :
  ∀ (X : Cᵒᵖ), m.app X = Hom.app { app := fun (X : Cᵒᵖ) ↦ (hc X).lift ((evaluation R X).mapCone s), naturality := ⋯ } X
⊢ m = { app := fun (X : Cᵒᵖ) ↦ (hc X).lift ((evaluation R X).mapCone s), naturality := ⋯ }

Tactic:
ext1 X
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
X : Cᵒᵖ
h : ∀ (j : J), m.app X ≫ ((evaluation R X).mapCone c).π.app j = ((evaluation R X).mapCone s).π.app j
⊢ m.app X = Hom.app { app := fun (X : Cᵒᵖ) ↦ (hc X).lift ((evaluation R X).mapCone s), naturality := ⋯ } X

Tactic:
apply (hc X).uniq ((evaluation R X).mapCone s)

Premises used: PresheafOfModules.evaluation, CategoryTheory.Limits.IsLimit.uniq, CategoryTheory.Functor.mapCone
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
X : Cᵒᵖ
j : J
h : m.app X ≫ (c.π.app j).app X = (s.π.app j).app X
⊢ m.app X ≫ ((evaluation R X).mapCone c).π.app j = ((evaluation R X).mapCone s).π.app j

Tactic:
dsimp
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
X : Cᵒᵖ
j : J
h : m.app X ≫ (c.π.app j).app X = (m ≫ c.π.app j).app X
h₁ h₂ : m.app X ≫ (c.π.app j).app X = m.app X ≫ (c.π.app j).app X
⊢ m.app X ≫ (c.π.app j).app X = (s.π.app j).app X

Tactic:
rw [← hm, comp_app]

Premises used: PresheafOfModules.comp_app
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
X Y : Cᵒᵖ
f : X ⟶ Y
change : HasLimit ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f)))
⊢ HasLimit (F ⋙ evaluation R Y ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f)))

Tactic:
change HasLimit ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (R.map f).hom)

Premises used: PresheafOfModules.evaluation, CategoryTheory.Functor.comp, CategoryTheory.Limits.HasLimit, ModuleCat.restrictScalars, CategoryTheory.Functor.map, RingCat.Hom.hom
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
X Y : Cᵒᵖ
f : X ⟶ Y
⊢ HasLimit ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f)))

Tactic:
infer_instance
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
X : Cᵒᵖ
rw :
  Eq (α :=
    limit (F ⋙ evaluation R X) ⟶
      limit ((F ⋙ evaluation R X) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
    (limMap (F.whiskerLeft (restriction R (𝟙 X))) :
      limit (F ⋙ evaluation R X) ⟶
        limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
    (Iso.inv (ModuleCat.restrictScalarsId'App (RingCat.Hom.hom (R.map (𝟙 X))) ⋯ (limit (F ⋙ evaluation R X))) ≫
        Iso.hom (preservesLimitIso (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) (F ⋙ evaluation R X)) :
      limit (F ⋙ evaluation R X) ⟶
        limit ((F ⋙ evaluation R X) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
⊢ Eq (α :=
    limit (F ⋙ evaluation R X) ⟶
      (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))).obj (limit (F ⋙ evaluation R X)))
    (CategoryStruct.comp (Y := limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
      (limMap (F.whiskerLeft (restriction R (𝟙 X))) :
        limit (F ⋙ evaluation R X) ⟶
          limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
      (Iso.inv (preservesLimitIso (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) (F ⋙ evaluation R X)) :
        limit ((F ⋙ evaluation R X) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) ⟶
          (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))).obj (limit (F ⋙ evaluation R X))))
    (Iso.inv (ModuleCat.restrictScalarsId'App (RingCat.Hom.hom (R.map (𝟙 X))) ⋯ (limit (F ⋙ evaluation R X))))

Tactic:
rw [← cancel_mono (preservesLimitIso _ _).hom, assoc, Iso.inv_hom_id, comp_id]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.inv_hom_id, CategoryTheory.cancel_mono, CategoryTheory.Category.comp_id, CategoryTheory.preservesLimitIso, CategoryTheory.Iso.hom
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
X : Cᵒᵖ
w :
  ∀ (j : J),
    Eq (α :=
      limit (F ⋙ evaluation R X) ⟶
        (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))).obj j)
      (CategoryStruct.comp (Y := limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
        (limMap (F.whiskerLeft (restriction R (𝟙 X))))
        (limit.π (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) j))
      (CategoryStruct.comp (Y := limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
        (Iso.inv (ModuleCat.restrictScalarsId'App (RingCat.Hom.hom (R.map (𝟙 X))) ⋯ (limit (F ⋙ evaluation R X))) ≫
            Iso.hom
              (preservesLimitIso (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) (F ⋙ evaluation R X)) :
          limit (F ⋙ evaluation R X) ⟶
            limit ((F ⋙ evaluation R X) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
        (limit.π (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) j :
          limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) ⟶
            (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))).obj j))
⊢ Eq (α :=
    limit (F ⋙ evaluation R X) ⟶
      limit ((F ⋙ evaluation R X) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
    (limMap (F.whiskerLeft (restriction R (𝟙 X))) :
      limit (F ⋙ evaluation R X) ⟶
        limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
    (Iso.inv (ModuleCat.restrictScalarsId'App (RingCat.Hom.hom (R.map (𝟙 X))) ⋯ (limit (F ⋙ evaluation R X))) ≫
        Iso.hom (preservesLimitIso (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) (F ⋙ evaluation R X)) :
      limit (F ⋙ evaluation R X) ⟶
        limit ((F ⋙ evaluation R X) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))

Tactic:
apply limit.hom_ext

Premises used: CategoryTheory.Limits.limit.hom_ext
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
X : Cᵒᵖ
⊢ ∀ (j : J),
    Eq (α :=
      limit (F ⋙ evaluation R X) ⟶
        (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))).obj j)
      (CategoryStruct.comp (Y := limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
        (limMap (F.whiskerLeft (restriction R (𝟙 X))))
        (limit.π (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) j))
      (CategoryStruct.comp (Y := limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
        (Iso.inv (ModuleCat.restrictScalarsId'App (RingCat.Hom.hom (R.map (𝟙 X))) ⋯ (limit (F ⋙ evaluation R X))) ≫
            Iso.hom
              (preservesLimitIso (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) (F ⋙ evaluation R X)) :
          limit (F ⋙ evaluation R X) ⟶
            limit ((F ⋙ evaluation R X) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))))
        (limit.π (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) j :
          limit (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))) ⟶
            (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (𝟙 X)))).obj j))

Tactic:
simp [← Functor.assoc, ← ModuleCat.restrictScalarsId'App_inv_naturality,
      ModuleCat.restrictScalarsId'_inv_app]

Premises used: CategoryTheory.Functor.assoc, ModuleCat.restrictScalarsId'App_inv_naturality, ModuleCat.restrictScalarsId'_inv_app
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
j : J
⊢ Eq (α :=
    limit (F ⋙ evaluation R X) ⟶
      Functor.obj ((F ⋙ evaluation R Z) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (f ≫ g)))) j)
    (CategoryStruct.comp (Y :=
      limit ((F ⋙ evaluation R Z) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (f ≫ g)))))
      (limMap (F.whiskerLeft (restriction R (f ≫ g))))
      (limit.π ((F ⋙ evaluation R Z) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (f ≫ g)))) j))
    (CategoryStruct.comp (Y := limit ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))))
      (limMap (F.whiskerLeft (restriction R f)) :
        limit (F ⋙ evaluation R X) ⟶
          limit ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))))
      (CategoryStruct.comp (Y :=
          (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).obj (limit (F ⋙ evaluation R Y)))
          (Iso.inv (preservesLimitIso (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))) (F ⋙ evaluation R Y)))
          ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map
              (CategoryStruct.comp (Y :=
                limit ((F ⋙ evaluation R Z) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map g))))
                (limMap (F.whiskerLeft (restriction R g)))
                (limit.π ((F ⋙ evaluation R Z) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map g))) j)) ≫
            Iso.inv
              (ModuleCat.restrictScalarsComp'App (RingCat.Hom.hom (R.map f)) (RingCat.Hom.hom (R.map g))
                (RingCat.Hom.hom (R.map (f ≫ g))) ⋯ ((F ⋙ evaluation R Z).obj j))) :
        limit ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))) ⟶
          Functor.obj ((F ⋙ evaluation R Z) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map (f ≫ g)))) j))

Tactic:
simp
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
j : J
X Y : Cᵒᵖ
f : X ⟶ Y
simp :
  Eq (α := limit (F ⋙ evaluation R X) ⟶ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).obj ((F.obj j).obj Y))
    (CategoryStruct.comp (Y := limit (F ⋙ evaluation R Y ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))))
      (limMap (F.whiskerLeft (restriction R f)))
      (limit.π ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))) j))
    (limit.π (F ⋙ evaluation R X) j ≫ (F.obj j).map f)
⊢ Eq (α := limit (F ⋙ evaluation R X) ⟶ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).obj ((F.obj j).obj Y))
    (CategoryStruct.comp (Y := (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).obj (limit (F ⋙ evaluation R Y)))
      (CategoryStruct.comp (Y := limit (F ⋙ evaluation R Y ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))))
        (limMap (F.whiskerLeft (restriction R f)) :
          limit (F ⋙ evaluation R X) ⟶
            limit (F ⋙ evaluation R Y ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))))
        (Iso.inv (preservesLimitIso (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))) (F ⋙ evaluation R Y)) :
          limit ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))) ⟶
            (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).obj (limit (F ⋙ evaluation R Y))))
      ((ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map (limit.π (F ⋙ evaluation R Y) j)))
    (limit.π (F ⋙ evaluation R X) j ≫ (F.obj j).map f)

Tactic:
simp only [assoc, preservesLimitIso_inv_π]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.preservesLimitIso_inv_π
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
j : J
X Y : Cᵒᵖ
f : X ⟶ Y
⊢ Eq (α := limit (F ⋙ evaluation R X) ⟶ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).obj ((F.obj j).obj Y))
    (CategoryStruct.comp (Y := limit (F ⋙ evaluation R Y ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))))
      (limMap (F.whiskerLeft (restriction R f)))
      (limit.π ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))) j))
    (limit.π (F ⋙ evaluation R X) j ≫ (F.obj j).map f)

Tactic:
apply limMap_π

Premises used: CategoryTheory.Limits.limMap_π
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
j j' : J
f : j ⟶ j'
h :
  ∀ (X : Cᵒᵖ),
    Eq (α := (Functor.obj (Functor.obj (Functor.const J) (limitPresheafOfModules F)) j).obj X ⟶ (F.obj j').obj X)
      (Hom.app
        (CategoryTheory.Functor.map (Functor.obj (Functor.const J) (limitPresheafOfModules F)) f ≫
          { app := fun (X : Cᵒᵖ) ↦ limit.π (F ⋙ evaluation R X) j', naturality := ⋯ })
        X)
      (Hom.app ({ app := fun (X : Cᵒᵖ) ↦ limit.π (F ⋙ evaluation R X) j, naturality := ⋯ } ≫ F.map f) X)
⊢ Eq (α := Functor.obj (Functor.obj (Functor.const J) (limitPresheafOfModules F)) j ⟶ F.obj j')
    (CategoryTheory.Functor.map (Functor.obj (Functor.const J) (limitPresheafOfModules F)) f ≫
      { app := fun (X : Cᵒᵖ) ↦ limit.π (F ⋙ evaluation R X) j', naturality := ⋯ })
    ({ app := fun (X : Cᵒᵖ) ↦ limit.π (F ⋙ evaluation R X) j, naturality := ⋯ } ≫ F.map f)

Tactic:
ext1 X
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits
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))
j j' : J
f : j ⟶ j'
X : Cᵒᵖ
⊢ Eq (α := (Functor.obj (Functor.obj (Functor.const J) (limitPresheafOfModules F)) j).obj X ⟶ (F.obj j').obj X)
    (Hom.app
      (CategoryTheory.Functor.map (Functor.obj (Functor.const J) (limitPresheafOfModules F)) f ≫
        { app := fun (X : Cᵒᵖ) ↦ limit.π (F ⋙ evaluation R X) j', naturality := ⋯ })
      X)
    (Hom.app ({ app := fun (X : Cᵒᵖ) ↦ limit.π (F ⋙ evaluation R X) j, naturality := ⋯ } ≫ F.map f) X)

Tactic:
simpa using (limit.w (F ⋙ evaluation R X) f).symm

Premises used: PresheafOfModules.evaluation, CategoryTheory.Functor.comp, Eq.symm, CategoryTheory.Limits.limit.w