-- Mathlib/CategoryTheory/Limits/FilteredColimitCommutesProduct.lean
theorem Types.isIso_colimitPointwiseProductToProductColimit (F : ∀ i, I i ⥤ Type u) :
    IsIso (colimitPointwiseProductToProductColimit F)

Proof state:
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.leftToMax ky ky') yk₀
yk' : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀'
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = colimit.ι (pointwiseProduct F) k yk
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = colimit.ι (pointwiseProduct F) k yk'
hy :
  colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk) =
    colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk')
s : α
hy₁ :
  Pi.π (fun s => colimit (F s)) s (colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk)) =
    colimit.ι (F s) (k s) (Pi.π (fun s => (F s).obj (k s)) s yk)
hy₂ :
  ∃ k_1 f g, (F s).map f (Pi.π (fun s => (F s).obj (k s)) s yk) = (F s).map g (Pi.π (fun s => (F s).obj (k s)) s yk')
⊢ ∃ i' hi', (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')

Tactic:
obtain ⟨i₀, f₀, g₀, h₀⟩ := hy₂

Result:
case intro.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.leftToMax ky ky') yk₀
yk' : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀'
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = colimit.ι (pointwiseProduct F) k yk
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = colimit.ι (pointwiseProduct F) k yk'
hy :
  colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk) =
    colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk')
s : α
hy₁ :
  Pi.π (fun s => colimit (F s)) s (colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk)) =
    colimit.ι (F s) (k s) (Pi.π (fun s => (F s).obj (k s)) s yk)
i₀ : I s
f₀ g₀ : k s ⟶ i₀
h₀ : (F s).map f₀ (Pi.π (fun s => (F s).obj (k s)) s yk) = (F s).map g₀ (Pi.π (fun s => (F s).obj (k s)) s yk')
⊢ ∃ i' hi', (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')

Proof state:
case refine_1.intro.intro.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
y' : colimit (pointwiseProduct F)
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = y'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : (pointwiseProduct F).obj k := (pointwiseProduct F).map (IsFiltered.leftToMax ky ky') yk₀
yk' : (pointwiseProduct F).obj k := (pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀'
hy :
  colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk) =
    colimitPointwiseProductToProductColimit F y'
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = colimit.ι (pointwiseProduct F) k yk
⊢ colimit.ι (pointwiseProduct F) k yk = y'

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

Result:
case refine_1.intro.intro.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : (pointwiseProduct F).obj k := (pointwiseProduct F).map (IsFiltered.leftToMax ky ky') yk₀
yk' : (pointwiseProduct F).obj k := (pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀'
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = colimit.ι (pointwiseProduct F) k yk
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = colimit.ι (pointwiseProduct F) k yk'
hy :
  colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk) =
    colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk')
⊢ colimit.ι (pointwiseProduct F) k yk = colimit.ι (pointwiseProduct F) k yk'

Proof state:
case refine_2
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
x : ∏ᶜ fun s => colimit (F s)
k : (s : α) → I s
p : (s : α) → (F s).obj (k s)
hk : ∀ (s : α), colimit.ι (F s) (k s) (p s) = Pi.π (fun s => colimit (F s)) s x
⊢ ∃ a, colimitPointwiseProductToProductColimit F a = x

Tactic:
refine ⟨colimit.ι (pointwiseProduct F) k ((Types.productIso _).inv p), ?_⟩

Result:
case refine_2
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
x : ∏ᶜ fun s => colimit (F s)
k : (s : α) → I s
p : (s : α) → (F s).obj (k s)
hk : ∀ (s : α), colimit.ι (F s) (k s) (p s) = Pi.π (fun s => colimit (F s)) s x
⊢ colimitPointwiseProductToProductColimit F
      (colimit.ι (pointwiseProduct F) k ((productIso fun s => (F s).obj (k s)).inv p)) =
    x

Proof state:
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
⊢ IsIso (colimitPointwiseProductToProductColimit F)

Tactic:
refine (isIso_iff_bijective _).2 ⟨fun y y' hy => ?_, fun x => ?_⟩

Result:
case refine_1
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
y y' : colimit (pointwiseProduct F)
hy : colimitPointwiseProductToProductColimit F y = colimitPointwiseProductToProductColimit F y'
⊢ y = y'

case refine_2
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
x : ∏ᶜ fun s => colimit (F s)
⊢ ∃ a, colimitPointwiseProductToProductColimit F a = x