-- Module: Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct
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)))
k' : (s : α) → I s
f : (s : α) → k s ⟶ k' s
hk' :
  ∀ (s : α),
    Eq (α := (fun (X : Type u) ↦ X) ((F s).obj (k' s)))
      ((hom (FC := TypeCat.Fun) ((F s).map (f s)) :
            (a : (fun (X : Type u) ↦ X) ((F s).obj (k s))) → (fun (X : Type u) ↦ X) ((F s).obj (k' s)))
          ((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 (k' s)))
      ((hom (FC := TypeCat.Fun) ((F s).map (f s)) :
            (a : (fun (X : Type u) ↦ X) ((F s).obj (k s))) → (fun (X : Type u) ↦ X) ((F s).obj (k' s)))
          ((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 (k' s)))
refine_1 :
  Eq (α := (fun (X : Type u) ↦ X) ((pointwiseProduct (I := I) F).obj k'))
    ((hom (FC := TypeCat.Fun) ((pointwiseProduct (I := I) F).map f) :
          (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj k)) →
            (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj k'))
        yk :
      (fun (X : Type u) ↦ X) ((pointwiseProduct (I := I) F).obj k'))
    ((hom (FC := TypeCat.Fun) ((pointwiseProduct (I := I) F).map f) :
          (a : (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj k)) →
            (fun (X : Type u) ↦ X) ((pointwiseProduct F).obj k'))
        yk' :
      (fun (X : Type u) ↦ X) ((pointwiseProduct (I := I) F).obj k'))
⊢ Eq (α := (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)))
    ((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)))

Tactic:
apply Types.colimit_sound' f f

Premises used: CategoryTheory.Limits.Types.colimit_sound'