-- Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.lean
/-- If a functor preserves equalizers and the appropriate products, it preserves limits. -/
lemma preservesLimit_of_preservesEqualizers_and_product :
    PreservesLimitsOfShape J G where
  preservesLimit {K}

Proof state:
C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ IsLimit (G.mapCone (buildLimit s t ⋯ ⋯ (limit.cone (parallelPair s t))))

Tactic:
apply IsLimit.ofIsoLimit (buildIsLimit _ _ _ _ _ _ _) _

Result:
C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ Fan (K ⋙ G).obj

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ Fan fun f => (K ⋙ G).obj f.fst.2

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ Cone.pt ?m.64686 ⟶ Cone.pt ?m.64687

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ Cone.pt ?m.64686 ⟶ Cone.pt ?m.64687

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ ∀ (f : (p : J × J) × (p.1 ⟶ p.2)),
    ?m.64688 ≫ (Cone.π ?m.64687).app { as := f } = (Cone.π ?m.64686).app { as := f.fst.1 } ≫ (K ⋙ G).map f.snd

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ ∀ (f : (p : J × J) × (p.1 ⟶ p.2)),
    ?m.64689 ≫ (Cone.π ?m.64687).app { as := f } = (Cone.π ?m.64686).app { as := f.fst.2 }

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ Fork ?m.64688 ?m.64689

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ IsLimit ?m.64686

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ IsLimit ?m.64687

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ IsLimit ?m.64692

C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ buildLimit ?m.64688 ?m.64689 ?m.64690 ?m.64691 ?m.64692 ≅
    G.mapCone (buildLimit s t ⋯ ⋯ (limit.cone (parallelPair s t)))