-- Module: Mathlib.CategoryTheory.Presentable.Limits
Goal:
C : Type u
inst : Category.{v, u} C
K : Type u'
inst_1 : Category.{v', u'} K
F : K ⥤ Functor.{v, w', u, w' + 1} C (Type w')
c : Cone F
hc : (Y : C) → IsLimit (F := F ⋙ obj (evaluation C (Type w')) Y) (mapCone (obj (evaluation C (Type w')) Y) c)
κ : Cardinal.{w}
inst_2 : Fact κ.IsRegular
hK : HasCardinalLT (Arrow.{v', u'} K) κ
J : Type w
inst_3 : SmallCategory J
inst_4 : IsCardinalFiltered J κ
X : Functor.{w, v, w, u} J C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
x : c.pt.obj cX.pt
this : IsFiltered J
y : (↑(F ⋙ obj (evaluation C (Type w')) cX.pt).sections : Type (max w' u'))
hy :
  (Equiv.symm (Types.isLimitEquivSections (hc cX.pt)) :
        (a : (↑(F ⋙ (evaluation C (Type w')).obj cX.pt).sections : Type (max w' u'))) →
          (((evaluation C (Type w')).obj cX.pt).mapCone c).pt)
      y =
    x
j₀ : J
z : (k : K) → (F.obj k).obj (X.obj j₀)
hz :
  ∀ (k : K),
    (↑y : (j : K) → (F ⋙ (evaluation C (Type w')).obj cX.pt).obj j) k =
      (ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k).map (cX.ι.app j₀)) :
          (a : (fun (X : Type w') ↦ X) ((F.obj k).obj (X.obj j₀))) →
            (fun (X : Type w') ↦ X) ((F.obj k).obj (((const J).obj cX.pt).obj j₀)))
        (z k)
j₁ : J
α : j₀ ⟶ j₁
hα :
  ∀ ⦃k k' : K⦄ (φ : k ⟶ k'),
    Eq (α := (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₁)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (X.map α)) :
            (a : (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀))) →
              (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₁)))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.map φ).app (X.obj j₀)) :
                (a : (fun (X : Type w') ↦ X) ((F.obj k).obj (X.obj j₀))) →
                  (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀)))
              (z k) :
            (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀))) :
        (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₁)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (X.map α)) :
            (a : (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀))) →
              (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₁)))
          (z k') :
        (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₁)))
k k' : K
φ : k ⟶ k'
rw :
  Eq (α := (F.obj k').obj (X.obj j₁))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (X.map α)) :
          (a : (F.obj k').obj (X.obj j₀)) → (F.obj k').obj (X.obj j₁))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.map φ).app (X.obj j₀)) :
              (a : (F.obj k).obj (X.obj j₀)) → (F.obj k').obj (X.obj j₀))
            (z k) :
          (F.obj k').obj (X.obj j₀)) :
      (F.obj k').obj (X.obj j₁))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (X.map α)) :
          (a : (F.obj k').obj (X.obj j₀)) → (F.obj k').obj (X.obj j₁))
        (z k') :
      (F.obj k').obj (X.obj j₁))
rw₁ rw₂ :
  Eq (α := (F.obj k').obj (X.obj j₁))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (X.map α)) :
          (a : (F.obj k').obj (X.obj j₀)) → (F.obj k').obj (X.obj j₁))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.map φ).app (X.obj j₀)) :
              (a : (F.obj k).obj (X.obj j₀)) → (F.obj k').obj (X.obj j₀))
            (z k) :
          (F.obj k').obj (X.obj j₀)) :
      (F.obj k').obj (X.obj j₁))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (X.map α)) :
          (a : (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀))) →
            (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₁)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.map φ).app (X.obj j₀)) :
              (a : (fun (X : Type w') ↦ X) ((F.obj k).obj (X.obj j₀))) →
                (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀)))
            (z k) :
          (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₀))) :
      (fun (X : Type w') ↦ X) ((F.obj k').obj (X.obj j₁)))
⊢ Eq (α := (F.obj k').obj (X.obj j₁))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.map φ).app (X.obj j₁)) :
          (a : (F.obj k).obj (X.obj j₁)) → (F.obj k').obj (X.obj j₁))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k).map (X.map α)) :
              (a : (F.obj k).obj (X.obj j₀)) → (F.obj k).obj (X.obj j₁))
            (z k) :
          (F.obj k).obj (X.obj j₁)) :
      (F.obj k').obj (X.obj j₁))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F.obj k').map (X.map α)) :
          (a : (F.obj k').obj (X.obj j₀)) → (F.obj k').obj (X.obj j₁))
        (z k') :
      (F.obj k').obj (X.obj j₁))

Tactic:
rw [NatTrans.naturality_apply, ← hα φ]

Premises used: CategoryTheory.NatTrans.naturality_apply