-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct (hypothesis)
Goal:
α : Type u
I : α → Type u
inst : (i : α) → SmallCategory (I i)
inst_1 : ∀ (i : α), IsFiltered.{u, u} (I i)
F : (i : α) → I i ⥤ Type u
ky : (i : α) → I i
yk₀ : (pointwiseProduct (I := I) F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct (I := I) F).obj ky'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : ∏ᶜ (Functor.pi (C := I) (D := fun (i : α) ↦ Type u) F).obj k :=
  (hom (FC := TypeCat.Fun) ((pointwiseProduct (I := I) F).map (IsFiltered.leftToMax ky ky')) :
      (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj ky)) →
        (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj (IsFiltered.max ky ky')))
    yk₀
yk' : ∏ᶜ (Functor.pi (C := I) (D := fun (i : α) ↦ Type u) F).obj k :=
  (hom (FC := TypeCat.Fun) ((pointwiseProduct (I := I) F).map (IsFiltered.rightToMax ky ky')) :
      (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj ky')) →
        (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj (IsFiltered.max ky ky')))
    yk₀'
hyk₀ :
  Eq (α := (fun (X : Type u) ↦ X) (colimit (pointwiseProduct (I := I) F)))
    ((hom (FC := TypeCat.Fun) (colimit.ι (pointwiseProduct (I := I) F) ky) :
          (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj ky)) →
            (fun (X : Type u) ↦ X) (colimit (pointwiseProduct F)))
        yk₀ :
      (fun (X : Type u) ↦ X) (colimit (pointwiseProduct (I := I) F)))
    ((hom (FC := TypeCat.Fun) (colimit.ι (pointwiseProduct (I := I) F) k) :
          (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj k)) →
            (fun (X : Type u) ↦ X) (colimit (pointwiseProduct F)))
        yk :
      (fun (X : Type u) ↦ X) (colimit (pointwiseProduct (I := I) F)))
hyk₀' :
  Eq (α := (fun (X : Type u) ↦ X) (colimit (pointwiseProduct (I := I) F)))
    ((hom (FC := TypeCat.Fun) (colimit.ι (pointwiseProduct (I := I) F) ky') :
          (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj ky')) →
            (fun (X : Type u) ↦ X) (colimit (pointwiseProduct F)))
        yk₀' :
      (fun (X : Type u) ↦ X) (colimit (pointwiseProduct (I := I) F)))
    ((hom (FC := TypeCat.Fun) (colimit.ι (pointwiseProduct (I := I) F) k) :
          (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj k)) →
            (fun (X : Type u) ↦ X) (colimit (pointwiseProduct F)))
        yk' :
      (fun (X : Type u) ↦ X) (colimit (pointwiseProduct (I := I) F)))
hy :
  Eq (α := (fun (X : Type u) ↦ X) (∏ᶜ fun (s : α) ↦ colimit (F s)))
    ((hom (FC := TypeCat.Fun) (colimitPointwiseProductToProductColimit (I := I) F) :
          (a : (fun (X : Type u) ↦ X) (colimit (pointwiseProduct F))) →
            (fun (X : Type u) ↦ X) (∏ᶜ fun (s : α) ↦ colimit (F s)))
        ((hom (FC := TypeCat.Fun) (colimit.ι (pointwiseProduct (I := I) F) k) :
              (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj k)) →
                (fun (X : Type u) ↦ X) (colimit (pointwiseProduct F)))
            yk :
          (fun (X : Type u) ↦ X) (colimit (pointwiseProduct (I := I) F))) :
      (fun (X : Type u) ↦ X) (∏ᶜ fun (s : α) ↦ colimit (F s)))
    ((hom (FC := TypeCat.Fun) (colimitPointwiseProductToProductColimit (I := I) F) :
          (a : (fun (X : Type u) ↦ X) (colimit (pointwiseProduct F))) →
            (fun (X : Type u) ↦ X) (∏ᶜ fun (s : α) ↦ colimit (F s)))
        ((hom (FC := TypeCat.Fun) (colimit.ι (pointwiseProduct (I := I) F) k) :
              (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj k)) →
                (fun (X : Type u) ↦ X) (colimit (pointwiseProduct F)))
            yk' :
          (fun (X : Type u) ↦ X) (colimit (pointwiseProduct (I := I) F))) :
      (fun (X : Type u) ↦ X) (∏ᶜ fun (s : α) ↦ colimit (F s)))
s : α
hy₁ :
  Eq (α := colimit (F s))
    ((hom (FC := TypeCat.Fun)
            (CategoryStruct.comp (Y := colimit (pointwiseProduct (I := I) F))
              (colimit.ι (pointwiseProduct (I := I) F) k)
              (colimitPointwiseProductToProductColimit (I := I) F ≫ Pi.π (fun (s : α) ↦ colimit (F s)) s)) :
          (a : (pointwiseProduct F).obj k) → colimit (F s))
        yk :
      colimit (F s))
    ((hom (FC := TypeCat.Fun)
            (Pi.π ((Functor.pi (C := I) (D := fun (i : α) ↦ Type u) F).obj k) s ≫ colimit.ι (F s) (k s)) :
          (a : (pointwiseProduct F).obj k) → colimit (F s))
        yk :
      colimit (F s))
⊢ ∃ (i' : I s) (hi' : k s ⟶ i'),
    Eq (α := (fun (X : Type u) ↦ X) ((F s).obj i'))
      ((hom (FC := TypeCat.Fun) ((F s).map hi') :
            (a : (fun (X : Type u) ↦ X) ((F s).obj (k s))) → (fun (X : Type u) ↦ X) ((F s).obj i'))
          ((hom (FC := TypeCat.Fun) (Pi.π (fun (s : α) ↦ (F s).obj (k s)) s) :
              (a : (fun (X : Type u) ↦ X) (∏ᶜ fun (s : α) ↦ (F s).obj (k s))) →
                (fun (X : Type u) ↦ X) ((F s).obj (k s)))
            yk) :
        (fun (X : Type u) ↦ X) ((F s).obj i'))
      ((hom (FC := TypeCat.Fun) ((F s).map hi') :
            (a : (fun (X : Type u) ↦ X) ((F s).obj (k s))) → (fun (X : Type u) ↦ X) ((F s).obj i'))
          ((hom (FC := TypeCat.Fun) (Pi.π (fun (s : α) ↦ (F s).obj (k s)) s) :
              (a : (fun (X : Type u) ↦ X) (∏ᶜ fun (s : α) ↦ (F s).obj (k s))) →
                (fun (X : Type u) ↦ X) ((F s).obj (k s)))
            yk') :
        (fun (X : Type u) ↦ X) ((F s).obj i'))
have hy₂ : Eq (α := colimit (F s))
  ((hom (FC := TypeCat.Fun)
          (CategoryStruct.comp (Y := colimit (pointwiseProduct (I := I) F)) (colimit.ι (pointwiseProduct (I := I) F) k)
            (colimitPointwiseProductToProductColimit (I := I) F ≫ Pi.π (fun (s : α) ↦ colimit (F s)) s)) :
        (a : (pointwiseProduct F).obj k) → colimit (F s))
      yk' :
    colimit (F s))
  ((hom (FC := TypeCat.Fun)
          (Pi.π ((Functor.pi (C := I) (D := fun (i : α) ↦ Type u) F).obj k) s ≫ colimit.ι (F s) (k s)) :
        (a : (pointwiseProduct F).obj k) → colimit (F s))
      yk' :
    colimit (F s))
Tactic: have hy₂ := congr_hom (ι_colimitPointwiseProductToProductColimit_π F k s) yk'