-- 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)))
⊢ ∀ (s : α),
    ∃ (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'))

Tactic:
have hch : ∀ (s : α), ∃ (i' : I s) (hi' : k s ⟶ i'),
        (F s).map hi' (Pi.π (fun s => (F s).obj (k s)) s yk) =
          (F s).map hi' (Pi.π (fun s => (F s).obj (k s)) s yk') := by
      intro s
      have hy₁ := congr_hom (ι_colimitPointwiseProductToProductColimit_π F k s) yk
      have hy₂ := congr_hom (ι_colimitPointwiseProductToProductColimit_π F k s) yk'
      dsimp at hy₁ hy₂ hy
      rw [← hy, hy₁, Types.FilteredColimit.colimit_eq_iff] at hy₂
      obtain ⟨i₀, f₀, g₀, h₀⟩ := hy₂
      refine ⟨IsFiltered.coeq f₀ g₀, f₀ ≫ IsFiltered.coeqHom f₀ g₀, ?_⟩
      conv_rhs => rw [IsFiltered.coeq_condition]
      dsimp [Functor.pi] at h₀
      simp [h₀]