awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
p p' : J × K
f : p ⟶ p'
rcases :
  ∀ (j : J) (k : K) (f : (j, k) ⟶ p'),
    s.π.app p'.1 ≫ NatTrans.app (Cone.π (D.obj p'.1)) p'.2 =
      s.π.app (j, k).1 ≫
        NatTrans.app (Cone.π (D.obj (j, k).1)) (j, k).2 ≫ (F.map f.1).app (j, k).2 ≫ (F.obj p'.1).map f.2
⊢ s.π.app p'.1 ≫ NatTrans.app (Cone.π (D.obj p'.1)) p'.2 =
    s.π.app p.1 ≫ NatTrans.app (Cone.π (D.obj p.1)) p.2 ≫ (F.map f.1).app p.2 ≫ (F.obj p'.1).map f.2

Tactic:
rcases p with ⟨j, k⟩
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
p' : J × K
j : J
k : K
f : (j, k) ⟶ p'
rcases :
  ∀ (j' : J) (k' : K) (f : (j, k) ⟶ (j', k')),
    s.π.app (j', k').1 ≫ NatTrans.app (Cone.π (D.obj (j', k').1)) (j', k').2 =
      s.π.app (j, k).1 ≫
        NatTrans.app (Cone.π (D.obj (j, k).1)) (j, k).2 ≫ (F.map f.1).app (j, k).2 ≫ (F.obj (j', k').1).map f.2
⊢ s.π.app p'.1 ≫ NatTrans.app (Cone.π (D.obj p'.1)) p'.2 =
    s.π.app (j, k).1 ≫ NatTrans.app (Cone.π (D.obj (j, k).1)) (j, k).2 ≫ (F.map f.1).app (j, k).2 ≫ (F.obj p'.1).map f.2

Tactic:
rcases p' with ⟨j', k'⟩
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
f : (j, k) ⟶ (j', k')
rcases :
  ∀ (fj : (j, k).1 ⟶ (j', k').1) (fk : (j, k).2 ⟶ (j', k').2),
    s.π.app (j', k').1 ≫ NatTrans.app (Cone.π (D.obj (j', k').1)) (j', k').2 =
      s.π.app (j, k).1 ≫
        NatTrans.app (Cone.π (D.obj (j, k).1)) (j, k).2 ≫
          (F.map (fj, fk).1).app (j, k).2 ≫ (F.obj (j', k').1).map (fj, fk).2
⊢ s.π.app (j', k').1 ≫ NatTrans.app (Cone.π (D.obj (j', k').1)) (j', k').2 =
    s.π.app (j, k).1 ≫
      NatTrans.app (Cone.π (D.obj (j, k).1)) (j, k).2 ≫ (F.map f.1).app (j, k).2 ≫ (F.obj (j', k').1).map f.2

Tactic:
rcases f with ⟨fj, fk⟩
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
dsimp : s.π.app j' ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.obj j).π.app k ≫ (F.map fj).app k ≫ (F.obj j').map fk
⊢ s.π.app (j', k').1 ≫ NatTrans.app (Cone.π (D.obj (j', k').1)) (j', k').2 =
    s.π.app (j, k).1 ≫
      NatTrans.app (Cone.π (D.obj (j, k).1)) (j, k).2 ≫
        (F.map (fj, fk).1).app (j, k).2 ≫ (F.obj (j', k').1).map (fj, fk).2

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
a : (F.obj j).map fk ≫ (F.map fj).app k'
| (F.map fj).app k ≫ (F.obj j').map fk

Tactic:
rw [← NatTrans.naturality]

Premises used: CategoryTheory.NatTrans.naturality
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
slice_rhs : s.π.app j' ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.obj j).π.app k ≫ (F.obj j).map fk ≫ (F.map fj).app k'
⊢ s.π.app j' ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.obj j).π.app k ≫ (F.map fj).app k ≫ (F.obj j').map fk

Tactic:
slice_rhs 3 4 => rw [← NatTrans.naturality]

Premises used: CategoryTheory.NatTrans.naturality
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
a : Functor.map (obj (const K) (D.obj j).pt) fk ≫ (D.obj j).π.app k'
| (D.obj j).π.app k ≫ (F.obj j).map fk

Tactic:
rw [← (D.obj j).π.naturality]

Premises used: CategoryTheory.Limits.Cone.π, CategoryTheory.Limits.DiagramOfCones.obj, CategoryTheory.NatTrans.naturality
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
slice_rhs :
  s.π.app j' ≫ (D.obj j').π.app k' =
    s.π.app j ≫ (Functor.map (obj (const K) (D.obj j).pt) fk ≫ (D.obj j).π.app k') ≫ (F.map fj).app k'
⊢ s.π.app j' ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.obj j).π.app k ≫ (F.obj j).map fk ≫ (F.map fj).app k'

Tactic:
slice_rhs 2 3 => rw [← (D.obj j).π.naturality]

Premises used: CategoryTheory.Limits.Cone.π, CategoryTheory.Limits.DiagramOfCones.obj, CategoryTheory.NatTrans.naturality
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
simp : s.π.app j' ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.obj j).π.app k' ≫ (F.map fj).app k'
⊢ s.π.app j' ≫ (D.obj j').π.app k' =
    s.π.app j ≫ (Functor.map (obj (const K) (D.obj j).pt) fk ≫ (D.obj j).π.app k') ≫ (F.map fj).app k'

Tactic:
simp only [Functor.const_obj_map, Category.id_comp, Category.assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Functor.const_obj_map, CategoryTheory.Category.id_comp
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
⊢ (D.map fj).hom ≫ NatTrans.app (Cone.π (D.obj (j', k').1)) k' =
    ((Cone.postcompose (F.map fj)).obj (D.obj (j, k).1)).π.app k'

Tactic:
have w := (D.map fj).w k'
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
w : (D.map fj).hom ≫ (D.obj j').π.app k' = (D.obj j).π.app k' ≫ (F.map fj).app k'
rw : s.π.app j' ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.map fj).hom ≫ (D.obj j').π.app k'
⊢ s.π.app j' ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.obj j).π.app k' ≫ (F.map fj).app k'

Tactic:
rw [← w]
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
w : (D.map fj).hom ≫ (D.obj j').π.app k' = (D.obj j).π.app k' ≫ (F.map fj).app k'
⊢ Functor.map (obj (const J) s.pt) fj ≫ s.π.app (j', k').1 = s.π.app (j, k).1 ≫ D.conePoints.map fj

Tactic:
have n := s.π.naturality fj
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
w : (D.map fj).hom ≫ (D.obj j').π.app k' = (D.obj j).π.app k' ≫ (F.map fj).app k'
n : 𝟙 s.pt ≫ s.π.app j' = s.π.app j ≫ (D.map fj).hom
⊢ s.π.app j' = s.π.app j ≫ (D.map fj).hom

Tactic:
simp only [Category.id_comp] at n

Premises used: CategoryTheory.Category.id_comp
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
w : (D.map fj).hom ≫ (D.obj j').π.app k' = (D.obj j).π.app k' ≫ (F.map fj).app k'
n : s.π.app j' = s.π.app j ≫ (D.map fj).hom
rw : (s.π.app j ≫ (D.map fj).hom) ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.map fj).hom ≫ (D.obj j').π.app k'
⊢ s.π.app j' ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.map fj).hom ≫ (D.obj j').π.app k'

Tactic:
rw [n]
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
j' : J
k' : K
fj : (j, k).1 ⟶ (j', k').1
fk : (j, k).2 ⟶ (j', k').2
w : (D.map fj).hom ≫ (D.obj j').π.app k' = (D.obj j).π.app k' ≫ (F.map fj).app k'
n : s.π.app j' = s.π.app j ≫ (D.map fj).hom
⊢ (s.π.app j ≫ (D.map fj).hom) ≫ (D.obj j').π.app k' = s.π.app j ≫ (D.map fj).hom ≫ (D.obj j').π.app k'

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
apply :
  ∀ (j_1 : K),
    (P.lift
            { pt := s.pt,
              π := { app := fun (p : J × K) ↦ s.π.app p.1 ≫ NatTrans.app (Cone.π (D.obj p.1)) p.2, naturality := ⋯ } } ≫
          (coneOfConeUncurry Q c).π.app j) ≫
        (D.obj j).π.app j_1 =
      s.π.app j ≫ (D.obj j).π.app j_1
⊢ P.lift
        { pt := s.pt,
          π := { app := fun (p : J × K) ↦ s.π.app p.1 ≫ NatTrans.app (Cone.π (D.obj p.1)) p.2, naturality := ⋯ } } ≫
      (coneOfConeUncurry Q c).π.app j =
    s.π.app j

Tactic:
apply (Q j).hom_ext

Premises used: CategoryTheory.Limits.IsLimit.hom_ext
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
j : J
k : K
⊢ (P.lift
          { pt := s.pt,
            π := { app := fun (p : J × K) ↦ s.π.app p.1 ≫ NatTrans.app (Cone.π (D.obj p.1)) p.2, naturality := ⋯ } } ≫
        (coneOfConeUncurry Q c).π.app j) ≫
      (D.obj j).π.app k =
    s.π.app j ≫ (D.obj j).π.app k

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
m : s.pt ⟶ (coneOfConeUncurry Q c).pt
w : ∀ (j : J), m ≫ (coneOfConeUncurry Q c).π.app j = s.π.app j
⊢ m =
    P.lift
      { pt := s.pt,
        π := { app := fun (p : J × K) ↦ s.π.app p.1 ≫ NatTrans.app (Cone.π (D.obj p.1)) p.2, naturality := ⋯ } }

Tactic:
P.uniq
      { pt := s.pt
        π := _ } m ?_

Premises used: CategoryTheory.Limits.Cone.pt, CategoryTheory.Limits.DiagramOfCones.conePoints, CategoryTheory.Limits.IsLimit.uniq, CategoryTheory.Limits.Cone.mk
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
m : s.pt ⟶ (coneOfConeUncurry Q c).pt
w : ∀ (j : J), m ≫ (coneOfConeUncurry Q c).π.app j = s.π.app j
refine :
  ∀ (j : J × K),
    m ≫ c.π.app j =
      NatTrans.app
        (Cone.π
          { pt := s.pt,
            π := { app := fun (p : J × K) ↦ s.π.app p.1 ≫ NatTrans.app (Cone.π (D.obj p.1)) p.2, naturality := ⋯ } })
        j
⊢ m =
    P.lift
      { pt := s.pt,
        π := { app := fun (p : J × K) ↦ s.π.app p.1 ≫ NatTrans.app (Cone.π (D.obj p.1)) p.2, naturality := ⋯ } }

Tactic:
refine P.uniq
      { pt := s.pt
        π := _ } m ?_

Premises used: CategoryTheory.Limits.Cone.pt, CategoryTheory.Limits.IsLimit.uniq, CategoryTheory.Limits.Cone.mk
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
m : s.pt ⟶ (coneOfConeUncurry Q c).pt
w : ∀ (j : J), m ≫ (coneOfConeUncurry Q c).π.app j = s.π.app j
j : J
k : K
dsimp : m ≫ c.π.app (j, k) = s.π.app j ≫ (D.obj j).π.app k
⊢ Eq (α :=
    Cone.pt (J := J × K) (F := obj uncurry F)
        { pt := s.pt,
          π :=
            { app := fun (p : J × K) ↦ s.π.app p.1 ≫ NatTrans.app (Cone.π (D.obj p.1)) p.2, naturality := ⋯ :
              NatTrans (obj (const (J × K)) s.pt) (obj uncurry F) } :
          Cone (obj uncurry F) } ⟶
      (obj uncurry F).obj (j, k))
    (m ≫ c.π.app (j, k))
    (NatTrans.app
      (Cone.π
        { pt := s.pt,
          π := { app := fun (p : J × K) ↦ s.π.app p.1 ≫ NatTrans.app (Cone.π (D.obj p.1)) p.2, naturality := ⋯ } })
      (j, k))

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
m : s.pt ⟶ (coneOfConeUncurry Q c).pt
w : ∀ (j : J), m ≫ (coneOfConeUncurry Q c).π.app j = s.π.app j
j : J
k : K
rw : m ≫ c.π.app (j, k) = (m ≫ (coneOfConeUncurry Q c).π.app j) ≫ (D.obj j).π.app k
⊢ m ≫ c.π.app (j, k) = s.π.app j ≫ (D.obj j).π.app k

Tactic:
rw [← w j]
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit c
s : Cone D.conePoints
m : s.pt ⟶ (coneOfConeUncurry Q c).pt
w : ∀ (j : J), m ≫ (coneOfConeUncurry Q c).π.app j = s.π.app j
j : J
k : K
⊢ m ≫ c.π.app (j, k) = (m ≫ (coneOfConeUncurry Q c).π.app j) ≫ (D.obj j).π.app k

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit (coneOfConeUncurry Q c)
E : (j : J) → Prod.sectR j K ⋙ obj uncurry F ≅ F.obj j :=
  fun (j : J) ↦ NatIso.ofComponents (fun (x : K) ↦ Iso.refl ((Prod.sectR j K ⋙ obj uncurry F).obj x)) ⋯
s : Cone (obj uncurry F)
j' j : J
f : j' ⟶ j
k : K
⊢ Eq (α := obj.{v_1, v_3, u_1, u_3} (obj (const J) s.pt) j' ⟶ (F.obj j).obj k)
    ((Functor.map (obj (const J) s.pt) f ≫
        (Q j).lift ((Cone.postcompose (E j).hom).obj (Cone.whisker (Prod.sectR j K) s))) ≫
      (D.obj j).π.app k)
    (((Q j').lift ((Cone.postcompose (E j').hom).obj (Cone.whisker (Prod.sectR j' K) s)) ≫ D.conePoints.map f) ≫
      (D.obj j).π.app k)

Tactic:
simpa [E] using s.π.naturality ((Prod.sectL J k).map f)

Premises used: CategoryTheory.Limits.Cone.π, CategoryTheory.Functor.map, CategoryTheory.Prod.sectL, CategoryTheory.NatTrans.naturality
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit (coneOfConeUncurry Q c)
E : (j : J) → Prod.sectR j K ⋙ obj uncurry F ≅ F.obj j :=
  fun (j : J) ↦ NatIso.ofComponents (fun (x : K) ↦ Iso.refl ((Prod.sectR j K ⋙ obj uncurry F).obj x)) ⋯
S : (s : Cone (obj uncurry F)) → Cone D.conePoints :=
  fun (s : Cone (obj uncurry F)) ↦
    { pt := s.pt,
      π :=
        { app := fun (j : J) ↦ (Q j).lift ((Cone.postcompose (E j).hom).obj (Cone.whisker (Prod.sectR j K) s)),
          naturality := ⋯ } }
s : Cone (obj uncurry F)
p : J × K
⊢ Eq (α :=
    Cone.pt (F := F.obj p.1) (obj (Cone.postcompose (Iso.hom (E p.1))) (Cone.whisker (Prod.sectR p.1 K) s)) ⟶
      (F.obj p.1).obj p.2)
    (lift (Q p.1) (obj (Cone.postcompose (Iso.hom (E p.1))) (Cone.whisker (Prod.sectR p.1 K) s)) ≫
      NatTrans.app (Cone.π (D.obj p.1)) p.2)
    (NatTrans.app (Cone.π (obj (Cone.postcompose (Iso.hom (E p.1))) (Cone.whisker (Prod.sectR p.1 K) s))) p.2)

Tactic:
have h1 := (Q p.1).fac ((Cone.postcompose (E p.1).hom).obj <|
        s.whisker (Prod.sectR p.1 K)) p.2
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit (coneOfConeUncurry Q c)
E : (j : J) → Prod.sectR j K ⋙ obj uncurry F ≅ F.obj j :=
  fun (j : J) ↦ NatIso.ofComponents (fun (x : K) ↦ Iso.refl ((Prod.sectR j K ⋙ obj uncurry F).obj x)) ⋯
S : (s : Cone (obj uncurry F)) → Cone D.conePoints :=
  fun (s : Cone (obj uncurry F)) ↦
    { pt := s.pt,
      π :=
        { app := fun (j : J) ↦ (Q j).lift ((Cone.postcompose (E j).hom).obj (Cone.whisker (Prod.sectR j K) s)),
          naturality := ⋯ } }
s : Cone (obj uncurry F)
p : J × K
h1 :
  Eq (α :=
    Cone.pt (F := F.obj p.1) (obj (Cone.postcompose (Iso.hom (E p.1))) (Cone.whisker (Prod.sectR p.1 K) s)) ⟶
      (F.obj p.1).obj p.2)
    (lift (Q p.1) (obj (Cone.postcompose (Iso.hom (E p.1))) (Cone.whisker (Prod.sectR p.1 K) s)) ≫
      NatTrans.app (Cone.π (D.obj p.1)) p.2)
    (NatTrans.app (Cone.π (obj (Cone.postcompose (Iso.hom (E p.1))) (Cone.whisker (Prod.sectR p.1 K) s))) p.2)
⊢ Eq (α := s.pt ⟶ (F.obj p.1).obj p.2)
    (CategoryStruct.comp (Y := Cone.pt (F := F.obj p.1) (D.obj p.1))
        (lift (Q p.1)
          (obj (C := Cone (Prod.sectR p.1 K ⋙ obj uncurry F))
            (Cone.postcompose (Iso.hom (NatIso.ofComponents (fun (x : K) ↦ Iso.refl ((F.obj p.1).obj x)) ⋯)))
            (Cone.whisker (Prod.sectR p.1 K) s)))
        (NatTrans.app (Cone.π (D.obj p.1)) p.2) :
      s.pt ⟶ (F.obj p.1).obj p.2)
    (s.π.app (p.1, p.2) : obj (obj (const (J × K)) s.pt) (p.1, p.2) ⟶ (obj uncurry F).obj (p.1, p.2))

Tactic:
simp only [Functor.comp_obj, Prod.sectR_obj, uncurry_obj_obj,
        Cone.postcompose_obj_pt, Cone.whisker_pt, Cone.postcompose_obj_π,
        Cone.whisker_π, NatTrans.comp_app, Functor.const_obj_obj, whiskerLeft_app,
        NatIso.ofComponents_hom_app, Iso.refl_hom, Category.comp_id, E] at h1

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Limits.Cone.whisker_π, CategoryTheory.Limits.Cone.postcompose_obj_pt, CategoryTheory.Functor.whiskerLeft_app, CategoryTheory.Limits.Cone.postcompose_obj_π, CategoryTheory.Functor.uncurry_obj_obj, CategoryTheory.Category.comp_id, CategoryTheory.Prod.sectR_obj, CategoryTheory.Limits.Cone.whisker_pt, CategoryTheory.Functor.const_obj_obj, CategoryTheory.NatTrans.comp_app, CategoryTheory.Iso.refl_hom, CategoryTheory.NatIso.ofComponents_hom_app
-- Module: Mathlib.CategoryTheory.Limits.Fubini
Goal:
J : Type u_1
K : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} K
C : Type u_3
inst_2 : Category.{v_3, u_3} C
F : J ⥤ Functor.{v_2, v_3, u_2, u_3} K C
G : J × K ⥤ C
D : DiagramOfCones F
Q : (j : J) → IsLimit (D.obj j)
c : Cone (obj uncurry F)
P : IsLimit (coneOfConeUncurry Q c)
E : (j : J) → Prod.sectR j K ⋙ obj uncurry F ≅ F.obj j :=
  fun (j : J) ↦ NatIso.ofComponents (fun (x : K) ↦ Iso.refl ((Prod.sectR j K ⋙ obj uncurry F).obj x)) ⋯
S : (s : Cone (obj uncurry F)) → Cone D.conePoints :=
  fun (s : Cone (obj uncurry F)) ↦
    { pt := s.pt,
      π :=
        { app := fun (j : J) ↦ (Q j).lift ((Cone.postcompose (E j).hom).obj (Cone.whisker (Prod.sectR j K) s)),
          naturality := ⋯ } }
s : Cone (obj uncurry F)
p : J × K
h1 :
  Eq (α := s.pt ⟶ (F.obj p.1).obj p.2)
    (CategoryStruct.comp (Y := Cone.pt (F := F.obj p.1) (D.obj p.1))
        (lift (Q p.1)
          (obj (C := Cone (Prod.sectR p.1 K ⋙ obj uncurry F))
            (Cone.postcompose (Iso.hom (NatIso.ofComponents (fun (x : K) ↦ Iso.refl ((F.obj p.1).obj x)) ⋯)))
            (Cone.whisker (Prod.sectR p.1 K) s)))
        (NatTrans.app (Cone.π (D.obj p.1)) p.2) :
      s.pt ⟶ (F.obj p.1).obj p.2)
    (s.π.app (p.1, p.2) : obj (obj (const (J × K)) s.pt) (p.1, p.2) ⟶ (obj uncurry F).obj (p.1, p.2))
⊢ Eq (α := (S s).pt ⟶ D.conePoints.obj p.1) (P.lift (S s) ≫ (coneOfConeUncurry Q c).π.app p.1) ((S s).π.app p.1)

Tactic:
have h2 := (P.fac (S s) p.1)