-- Mathlib/CategoryTheory/Filtered/Small.lean
theorem small_fullSubcategory_cofilteredClosure :
    Small.{max v w} (FullSubcategory (CofilteredClosure f))

Proof state:
case mk.eq.intro.mk.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩)
m : ℕ
y : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f m).fst
hj₂ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩)
g g' :
  CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩ ⟶
    CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩
⊢ ∃ a,
    CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      { obj := eq g g', property := ⋯ }.obj

Tactic:
refine ⟨⟨(Max.max n m).succ, CofilteredClosureSmall.InductiveStep.eq ?_ ?_ x y g g'⟩, rfl⟩

Result:
case mk.eq.intro.mk.intro.mk.refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩)
m : ℕ
y : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f m).fst
hj₂ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩)
g g' :
  CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩ ⟶
    CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩
⊢ n < n ⊔ m + 1

case mk.eq.intro.mk.intro.mk.refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩)
m : ℕ
y : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f m).fst
hj₂ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩)
g g' :
  CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩ ⟶
    CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩
⊢ m < n ⊔ m + 1

Proof state:
case mk.min.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j j'✝ : C
hj₂ : CofilteredClosure f j'✝
ih' :
  ∃ a,
    CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      { obj := j'✝, property := hj₂ }.obj
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩)
⊢ ∃ a,
    CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      {
          obj :=
            min (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩) j'✝,
          property := ⋯ }.obj

Tactic:
rcases ih' with ⟨⟨m, y⟩, rfl⟩

Result:
case mk.min.intro.mk.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩)
m : ℕ
y : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f m).fst
hj₂ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩)
⊢ ∃ a,
    CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
      {
          obj :=
            min (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩)
              (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩),
          property := ⋯ }.obj

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
⊢ Small.{max v w, u} (FullSubcategory (CofilteredClosure f))

Tactic:
refine small_of_injective_of_exists
  (CofilteredClosureSmall.abstractCofilteredClosureRealization f)
  (fun _ _ => FullSubcategory.ext) ?_

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
⊢ ∀ (b : FullSubcategory (CofilteredClosure f)),
    ∃ a, CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a = b.obj

Proof state:
case mk.eq.intro.mk.intro.mk.refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩)
m : ℕ
y : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f m).fst
hj₂ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩)
g g' :
  CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩ ⟶
    CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩
⊢ m < n ⊔ m + 1

Tactic:
apply Nat.lt_succ_of_le

Result:
case mk.eq.intro.mk.intro.mk.refine_2.a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩)
m : ℕ
y : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f m).fst
hj₂ :
  CofilteredClosure f (CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩)
g g' :
  CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨n, x⟩ ⟶
    CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f ⟨m, y⟩
⊢ m ≤ n ⊔ m