-- Module: Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit
Goal:
J : Type u₁
K : Type u₂
inst : SmallCategory J
inst_1 : Category.{v₂, u₂} K
inst_2 : Small.{v, u₂} K
inst_3 : FinCategory J
F : J × K ⥤ Type v
inst_4 : IsFiltered.{v₂, u₂} K
x : limit (obj curry F ⋙ colim)
k : J → K
y : (j : J) → F.obj (j, k j)
e :
  ∀ (j : J),
    Eq (α := (fun (X : Type v) ↦ X) (colimit ((obj curry F).obj j)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι ((obj curry F).obj j) (k j)) :
            (a : (fun (X : Type v) ↦ X) (((curry.obj F).obj j).obj (k j))) →
              (fun (X : Type v) ↦ X) (colimit ((curry.obj F).obj j)))
          (y j) :
        (fun (X : Type v) ↦ X) (colimit ((obj curry F).obj j)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (limit.π (obj curry F ⋙ colim) j) :
            (a : (fun (X : Type v) ↦ X) (limit (curry.obj F ⋙ colim))) →
              (fun (X : Type v) ↦ X) ((curry.obj F ⋙ colim).obj j))
          x :
        (fun (X : Type v) ↦ X) ((obj curry F ⋙ colim).obj j))
k' : K
g : (j : J) → k j ⟶ k'
kf : {j j' : J} → (j ⟶ j') → K
gf hf : {j j' : J} → (f : j ⟶ j') → k' ⟶ kf f
wf :
  ∀ {j j' : J} (f : j ⟶ j'),
    Eq (α := (fun (X : Type v) ↦ X) (F.obj (j', kf f)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (𝟙 j' ×ₘ g j' ≫ gf f)) :
            (a : (fun (X : Type v) ↦ X) (F.obj (j', k j'))) → (fun (X : Type v) ↦ X) (F.obj (j', kf f)))
          (y j') :
        (fun (X : Type v) ↦ X) (F.obj (j', kf f)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (f ×ₘ g j ≫ hf f)) :
            (a : (fun (X : Type v) ↦ X) (F.obj (j, k j))) → (fun (X : Type v) ↦ X) (F.obj (j', kf f)))
          (y j) :
        (fun (X : Type v) ↦ X) (F.obj (j', kf f)))
k'' : K
i : {j j' : J} → (f : j ⟶ j') → kf f ⟶ k''
s : ∀ {j₁ j₂ j₃ j₄ : J} (f : j₁ ⟶ j₂) (f' : j₃ ⟶ j₄), gf f ≫ i f = hf f' ≫ i f'
j : J
h :
  Eq (α := colimit ((obj curry F).obj j))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι ((obj curry F).obj j) k'') :
          (a : F.obj (j, k'')) → colimit ((curry.obj F).obj j))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (limit.π ((obj curry (swap K J ⋙ F)).obj k'') j) :
              (a : limit ((curry.obj (swap K J ⋙ F)).obj k'')) → F.obj (j, k''))
            (Limit.mk ((obj curry (swap K J ⋙ F)).obj k'')
              (fun (j : J) ↦
                (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (𝟙 j ×ₘ i (𝟙 j))) :
                    (a : F.obj (j, kf (𝟙 j))) → F.obj (j, k''))
                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (𝟙 j ×ₘ gf (𝟙 j))) :
                        (a : F.obj (j, k')) → F.obj (j, kf (𝟙 j)))
                      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (𝟙 j ×ₘ g j)) :
                            (a : F.obj (j, k j)) → F.obj (j, k'))
                          (y j) :
                        F.obj (j, k')) :
                    F.obj (j, kf (𝟙 j))))
              ⋯) :
          F.obj (j, k'')) :
      colimit ((obj curry F).obj j))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (limit.π (obj curry F ⋙ colim) j) :
          (a : limit (curry.obj F ⋙ colim)) → colimit ((curry.obj F).obj j))
        x :
      colimit ((obj curry F).obj j))
⊢ Eq (α := (fun (X : Type v) ↦ X) ((obj curry F ⋙ colim).obj j))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (limit.π (obj curry F ⋙ colim) j) :
          (a : (fun (X : Type v) ↦ X) (limit (curry.obj F ⋙ colim))) →
            (fun (X : Type v) ↦ X) ((curry.obj F ⋙ colim).obj j))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimitLimitToLimitColimit F) :
              (a : (fun (X : Type v) ↦ X) (colimit (curry.obj (swap K J ⋙ F) ⋙ lim))) →
                (fun (X : Type v) ↦ X) (limit (curry.obj F ⋙ colim)))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι (obj curry (swap K J ⋙ F) ⋙ lim) k'') :
                  (a : (fun (X : Type v) ↦ X) ((curry.obj (swap K J ⋙ F) ⋙ lim).obj k'')) →
                    (fun (X : Type v) ↦ X) (colimit (curry.obj (swap K J ⋙ F) ⋙ lim)))
                (id
                  (Limit.mk ((obj curry (swap K J ⋙ F)).obj k'')
                    (fun (j : J) ↦
                      (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (𝟙 j ×ₘ g j ≫ gf (𝟙 j) ≫ i (𝟙 j))) :
                          (a : (fun (X : Type v) ↦ X) (F.obj (j, k j))) → (fun (X : Type v) ↦ X) (F.obj (j, k'')))
                        (y j))
                    ⋯)) :
              (fun (X : Type v) ↦ X) (colimit (obj curry (swap K J ⋙ F) ⋙ lim))) :
          (fun (X : Type v) ↦ X) (limit (obj curry F ⋙ colim))) :
      (fun (X : Type v) ↦ X) ((obj curry F ⋙ colim).obj j))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (limit.π (obj curry F ⋙ colim) j) :
          (a : (fun (X : Type v) ↦ X) (limit (curry.obj F ⋙ colim))) →
            (fun (X : Type v) ↦ X) ((curry.obj F ⋙ colim).obj j))
        x :
      (fun (X : Type v) ↦ X) ((obj curry F ⋙ colim).obj j))

Tactic:
simp only [comp_obj, colim_obj, lim_obj, Bifunctor.map_id_comp, comp_apply, id_eq,
        ι_colimitLimitToLimitColimit_π_apply]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Bifunctor.map_id_comp, CategoryTheory.Limits.lim_obj, CategoryTheory.Limits.colim_obj, CategoryTheory.comp_apply, CategoryTheory.Limits.ι_colimitLimitToLimitColimit_π_apply, id_eq