-- 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
y' : (fun (X : Type u) ↦ X) (colimit (pointwiseProduct (I := I) F))
ky : (i : α) → I i
yk₀ : (pointwiseProduct (I := I) F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct (I := I) F).obj ky'
hyk₀' :
  (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' =
    y'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : (pointwiseProduct (I := I) 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' : (pointwiseProduct (I := I) 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'
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)))
        y' :
      (fun (X : Type u) ↦ X) (∏ᶜ fun (s : α) ↦ colimit (F s)))
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)))
refine_1 :
  ∀
    (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)))),
    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)))
⊢ (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 =
    y'

Tactic:
obtain rfl : y' = colimit.ι (pointwiseProduct F) k yk' := by
      simp only [k, yk', colimit.w_apply, hyk₀']

Premises used: CategoryTheory.ConcreteCategory.hom, CategoryTheory.Limits.pointwiseProduct, DFunLike.coe, rfl, CategoryTheory.Limits.colimit.ι