-- 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'
j j' : J
f : j ⟶ j'
simp :
  Eq (α := colimit ((obj curry F).obj j'))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι ((obj curry F).obj j') (k j')) :
          (a : ((curry.obj F).obj j').obj (k j')) → colimit ((curry.obj F).obj j'))
        (y j') :
      colimit ((obj curry F).obj j'))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι ((obj curry F).obj j') (k j)) :
          (a : ((curry.obj F).obj j').obj (k j)) → colimit ((curry.obj F).obj j'))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (f ×ₘ 𝟙 (k j))) :
              (a : ((curry.obj F).obj j).obj (k j)) → ((curry.obj F).obj j').obj (k j))
            (y j) :
          ((obj curry F).obj j').obj (k j)) :
      colimit ((obj curry F).obj 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') :
          (a : (fun (X : Type v) ↦ X) (((curry.obj F).obj j').obj k')) →
            (fun (X : Type v) ↦ X) (colimit ((curry.obj F).obj j')))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (𝟙 j' ×ₘ g 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) (F.obj (j', k'))) :
      (fun (X : Type v) ↦ X) (colimit ((obj curry F).obj j')))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι ((obj curry F).obj j') k') :
          (a : (fun (X : Type v) ↦ X) (((curry.obj F).obj j').obj k')) →
            (fun (X : Type v) ↦ X) (colimit ((curry.obj F).obj j')))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (f ×ₘ 𝟙 (k j)) ≫ F.map (𝟙 j' ×ₘ g 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) (F.obj (j', k'))) :
      (fun (X : Type v) ↦ X) (colimit ((obj curry F).obj j')))

Tactic:
simp only [← curry_obj_obj_map, ← curry_obj_obj_obj, comp_apply, colimit.w_apply]

Premises used: CategoryTheory.Functor.curry_obj_obj_obj, CategoryTheory.Limits.colimit.w_apply, CategoryTheory.comp_apply, CategoryTheory.Functor.curry_obj_obj_map