-- 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)
obtain :
  ∀ (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)))),
    ∃ (j : J) (x' : c.pt.obj (X.obj j)),
      x =
        (ConcreteCategory.hom (FC := TypeCat.Fun) ((c.pt.mapCocone cX).ι.app j) :
            (a : (fun (X : Type w') ↦ X) ((X ⋙ c.pt).obj j)) →
              (fun (X : Type w') ↦ X) (((const J).obj (c.pt.mapCocone cX).pt).obj j))
          x'
⊢ ∃ (j : J) (x' : c.pt.obj (X.obj j)),
    x =
      (ConcreteCategory.hom (FC := TypeCat.Fun) ((c.pt.mapCocone cX).ι.app j) :
          (a : (fun (X : Type w') ↦ X) ((X ⋙ c.pt).obj j)) →
            (fun (X : Type w') ↦ X) (((const J).obj (c.pt.mapCocone cX).pt).obj j))
        x'

Tactic:
obtain ⟨j₁, α, hα⟩ : ∃ (j₁ : J) (α : j₀ ⟶ j₁), ∀ ⦃k k' : K⦄ (φ : k ⟶ k'),
      (F.obj k').map (X.map α) ((F.map φ).app _ (z k)) =
        (F.obj k').map (X.map α) (z k') := by
    have H {k k' : K} (φ : k ⟶ k') :=
      (Types.FilteredColimit.isColimit_eq_iff' (ht := hF k')
        (x := (F.map φ).app _ (z k)) (y := z k')).1 (by
          dsimp at hz ⊢
          simpa only [← NatTrans.naturality_apply, ← hz] using y.2 φ)
    let j {k k' : K} (φ : k ⟶ k') : J := (H φ).choose
    let g {k k' : K} (φ : k ⟶ k') : j₀ ⟶ j φ := (H φ).choose_spec.choose
    have hg {k k' : K} (φ : k ⟶ k') :
        (F.obj k').map (X.map (g φ)) ((F.map φ).app _ (z k)) =
          (F.obj k').map (X.map (g φ)) (z k') := (H φ).choose_spec.choose_spec
    obtain ⟨j₁, α, β, hα⟩ : ∃ (j₁ : J) (α : j₀ ⟶ j₁)
        (β : ∀ ⦃k k' : K⦄ (φ : k ⟶ k'), j φ ⟶ j₁),
        ∀ ⦃k k' : K⦄ (φ : k ⟶ k'), α = g φ ≫ β φ := by
      let j'' (f : Arrow K) : J := j f.hom
      let ψ (f : Arrow K) : j₀ ⟶ IsCardinalFiltered.max j'' hK :=
        g f.hom ≫ IsCardinalFiltered.toMax j'' hK f
      refine ⟨IsCardinalFiltered.coeq ψ hK, IsCardinalFiltered.toCoeq ψ hK,
        fun k k' φ ↦ IsCardinalFiltered.toMax j'' hK φ ≫ IsCardinalFiltered.coeqHom ψ hK,
        fun k k' φ ↦ ?_⟩
      simpa [ψ] using (IsCardinalFiltered.coeq_condition ψ hK (Arrow.mk φ)).symm
    exact ⟨j₁, α, fun k k' φ ↦ by simp [hα φ, hg]⟩