-- 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))
h2 : 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)
dsimp :
  P.lift
        { pt := s.pt,
          π :=
            {
              app := fun (j : J) ↦
                (Q j).lift
                  (obj (C := Cone (Prod.sectR j K ⋙ obj uncurry F))
                    (Cone.postcompose (Iso.hom (NatIso.ofComponents (fun (x : K) ↦ Iso.refl ((F.obj j).obj x)) ⋯)))
                    (Cone.whisker (Prod.sectR j K) s)),
              naturality := ⋯ } } ≫
      c.π.app p =
    s.π.app p
⊢ P.lift (S s) ≫ c.π.app p = s.π.app p

Tactic:
dsimp only [Functor.comp_obj, Prod.sectR_obj, uncurry_obj_obj, NatTrans.id_app,
        Functor.const_obj_obj, DiagramOfCones.conePoints_obj, DiagramOfCones.conePoints_map,
        Functor.const_obj_map, id_eq, Cone.postcompose_obj_pt, Cone.whisker_pt,
        Cone.postcompose_obj_π, Cone.whisker_π, NatTrans.comp_app, whiskerLeft_app,
        NatIso.ofComponents_hom_app, Iso.refl_hom, Prod.sectL_obj, Prod.sectL_map, eq_mp_eq_cast,
        eq_mpr_eq_cast, coneOfConeUncurry_pt, coneOfConeUncurry_π_app, S, E] at h2 ⊢

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Prod.sectL_obj, CategoryTheory.Limits.Cone.whisker_π, CategoryTheory.Limits.Cone.postcompose_obj_pt, eq_mp_eq_cast, CategoryTheory.Functor.whiskerLeft_app, CategoryTheory.Limits.Cone.postcompose_obj_π, CategoryTheory.Functor.uncurry_obj_obj, CategoryTheory.Limits.DiagramOfCones.conePoints_map, CategoryTheory.Limits.DiagramOfCones.conePoints_obj, CategoryTheory.Limits.coneOfConeUncurry_π_app, CategoryTheory.Prod.sectR_obj, CategoryTheory.Prod.sectL_map, CategoryTheory.NatTrans.id_app, CategoryTheory.Limits.Cone.whisker_pt, eq_mpr_eq_cast, CategoryTheory.Limits.coneOfConeUncurry_pt, CategoryTheory.Functor.const_obj_obj, CategoryTheory.NatTrans.comp_app, CategoryTheory.Iso.refl_hom, CategoryTheory.Functor.const_obj_map, id_eq, CategoryTheory.NatIso.ofComponents_hom_app