-- Mathlib/CategoryTheory/Localization/SmallShiftedHom.lean
lemma hasSmallLocalizedShiftedHom_iff
    (L : C ⥤ D) [L.IsLocalization W] [L.CommShift M] (X Y : C) :
    HasSmallLocalizedShiftedHom.{w} W M X Y ↔
      ∀ (a b : M), Small.{w} ((L.obj X)⟦a⟧ ⟶ (L.obj Y)⟦b⟧)

Proof state:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
L : C ⥤ D
inst✝¹ : L.IsLocalization W
inst✝ : L.CommShift M
X Y : C
⊢ (∀ (a b : M), HasSmallLocalizedHom W ((shiftFunctor C a).obj X) ((shiftFunctor C b).obj Y)) ↔
    ∀ (a b : M), Small.{w, v₂} ((shiftFunctor D a).obj (L.obj X) ⟶ (shiftFunctor D b).obj (L.obj Y))

Tactic:
have eq := fun (a b : M) ↦ small_congr.{w}
  (Iso.homCongr ((L.commShiftIso a).app X) ((L.commShiftIso b).app Y))

Result:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
L : C ⥤ D
inst✝¹ : L.IsLocalization W
inst✝ : L.CommShift M
X Y : C
eq :
  ∀ (a b : M),
    Small.{w, v₂} ((shiftFunctor C a ⋙ L).obj X ⟶ (shiftFunctor C b ⋙ L).obj Y) ↔
      Small.{w, v₂} ((L ⋙ shiftFunctor D a).obj X ⟶ (L ⋙ shiftFunctor D b).obj Y)
⊢ (∀ (a b : M), HasSmallLocalizedHom W ((shiftFunctor C a).obj X) ((shiftFunctor C b).obj Y)) ↔
    ∀ (a b : M), Small.{w, v₂} ((shiftFunctor D a).obj (L.obj X) ⟶ (shiftFunctor D b).obj (L.obj Y))

Proof state:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
L : C ⥤ D
inst✝¹ : L.IsLocalization W
inst✝ : L.CommShift M
X Y : C
⊢ HasSmallLocalizedShiftedHom W M X Y ↔
    ∀ (a b : M), Small.{w, v₂} ((shiftFunctor D a).obj (L.obj X) ⟶ (shiftFunctor D b).obj (L.obj Y))

Tactic:
dsimp [HasSmallLocalizedShiftedHom]

Result:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
L : C ⥤ D
inst✝¹ : L.IsLocalization W
inst✝ : L.CommShift M
X Y : C
⊢ (∀ (a b : M), HasSmallLocalizedHom W ((shiftFunctor C a).obj X) ((shiftFunctor C b).obj Y)) ↔
    ∀ (a b : M), Small.{w, v₂} ((shiftFunctor D a).obj (L.obj X) ⟶ (shiftFunctor D b).obj (L.obj Y))

Proof state:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
L : C ⥤ D
inst✝¹ : L.IsLocalization W
inst✝ : L.CommShift M
X Y : C
eq :
  ∀ (a b : M),
    Small.{w, v₂} (L.obj ((shiftFunctor C a).obj X) ⟶ L.obj ((shiftFunctor C b).obj Y)) ↔
      Small.{w, v₂} ((shiftFunctor D a).obj (L.obj X) ⟶ (shiftFunctor D b).obj (L.obj Y))
⊢ (∀ (a b : M), HasSmallLocalizedHom W ((shiftFunctor C a).obj X) ((shiftFunctor C b).obj Y)) ↔
    ∀ (a b : M), Small.{w, v₂} ((shiftFunctor D a).obj (L.obj X) ⟶ (shiftFunctor D b).obj (L.obj Y))

Tactic:
simp only [hasSmallLocalizedHom_iff _ L, eq]

Result:
no goals

Proof state:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
L : C ⥤ D
inst✝¹ : L.IsLocalization W
inst✝ : L.CommShift M
X Y : C
eq :
  ∀ (a b : M),
    Small.{w, v₂} ((shiftFunctor C a ⋙ L).obj X ⟶ (shiftFunctor C b ⋙ L).obj Y) ↔
      Small.{w, v₂} ((L ⋙ shiftFunctor D a).obj X ⟶ (L ⋙ shiftFunctor D b).obj Y)
⊢ (∀ (a b : M), HasSmallLocalizedHom W ((shiftFunctor C a).obj X) ((shiftFunctor C b).obj Y)) ↔
    ∀ (a b : M), Small.{w, v₂} ((shiftFunctor D a).obj (L.obj X) ⟶ (shiftFunctor D b).obj (L.obj Y))

Tactic:
dsimp at eq

Result:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
L : C ⥤ D
inst✝¹ : L.IsLocalization W
inst✝ : L.CommShift M
X Y : C
eq :
  ∀ (a b : M),
    Small.{w, v₂} (L.obj ((shiftFunctor C a).obj X) ⟶ L.obj ((shiftFunctor C b).obj Y)) ↔
      Small.{w, v₂} ((shiftFunctor D a).obj (L.obj X) ⟶ (shiftFunctor D b).obj (L.obj Y))
⊢ (∀ (a b : M), HasSmallLocalizedHom W ((shiftFunctor C a).obj X) ((shiftFunctor C b).obj Y)) ↔
    ∀ (a b : M), Small.{w, v₂} ((shiftFunctor D a).obj (L.obj X) ⟶ (shiftFunctor D b).obj (L.obj Y))