ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Localization.Construction.morphismProperty_is_top
theorem morphismProperty_is_top (P : MorphismProperty W.Localization)
    [P.IsStableUnderComposition] (hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f))
    (hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)) :
    P = ⊤ 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] {W : CategoryTheory.MorphismProperty C} (P : CategoryTheory.MorphismProperty W.Localization) [P.IsStableUnderComposition] (hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)) (hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (CategoryTheory.Localization.Construction.wInv w hw))
Goal: P = ⊤
-- CategoryTheory.Localization.Construction.morphismProperty_is_top'
theorem morphismProperty_is_top' (P : MorphismProperty W.Localization)
    [P.IsStableUnderComposition] (hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f))
    (hP₂ : ∀ ⦃X Y : W.Localization⦄ (e : X ≅ Y) (_ : P e.hom), P e.inv) : P = ⊤ 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] {W : CategoryTheory.MorphismProperty C} (P : CategoryTheory.MorphismProperty W.Localization) [P.IsStableUnderComposition] (hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)) (hP₂ : ∀ ⦃X Y : W.Localization⦄ (e : X ≅ Y), P e.hom → P e.inv)
Goal: P = ⊤
-- CategoryTheory.Localization.Construction.NatTransExtension.app_eq
@[simp]
theorem app_eq (X : C) : (app τ) (W.Q.obj X) = τ.app X 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] {W : CategoryTheory.MorphismProperty C} {D : Type uD} [CategoryTheory.Category.{uD', uD} D] {F₁ F₂ : CategoryTheory.Functor W.Localization D} (τ : W.Q.comp F₁ ⟶ W.Q.comp F₂) (X : C)
Goal: CategoryTheory.Localization.Construction.NatTransExtension.app τ (W.Q.obj X) = τ.app X
-- CategoryTheory.Localization.Construction.natTransExtension_app
@[simps]
def natTransExtension {F₁ F₂ : W.Localization ⥤ D} (τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂) : F₁ ⟶ F₂ 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] {W : CategoryTheory.MorphismProperty C} {D : Type uD} [CategoryTheory.Category.{uD', uD} D] {F₁ F₂ : CategoryTheory.Functor W.Localization D} (τ : W.Q.comp F₁ ⟶ W.Q.comp F₂) (X : W.Localization)
Goal: (CategoryTheory.Localization.Construction.natTransExtension τ).app X =
  CategoryTheory.Localization.Construction.NatTransExtension.app τ X
-- CategoryTheory.Localization.Construction.natTransExtension_hcomp
@[simp]
theorem natTransExtension_hcomp {F G : W.Localization ⥤ D} (τ : W.Q ⋙ F ⟶ W.Q ⋙ G) :
    𝟙 W.Q ◫ natTransExtension τ = τ 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] {W : CategoryTheory.MorphismProperty C} {D : Type uD} [CategoryTheory.Category.{uD', uD} D] {F G : CategoryTheory.Functor W.Localization D} (τ : W.Q.comp F ⟶ W.Q.comp G)
Goal: CategoryTheory.CategoryStruct.id W.Q ◫ CategoryTheory.Localization.Construction.natTransExtension τ = τ
-- CategoryTheory.Localization.Construction.natTrans_hcomp_injective
theorem natTrans_hcomp_injective {F G : W.Localization ⥤ D} {τ₁ τ₂ : F ⟶ G}
    (h : 𝟙 W.Q ◫ τ₁ = 𝟙 W.Q ◫ τ₂) : τ₁ = τ₂ 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] {W : CategoryTheory.MorphismProperty C} {D : Type uD} [CategoryTheory.Category.{uD', uD} D] {F G : CategoryTheory.Functor W.Localization D} {τ₁ τ₂ : F ⟶ G} (h : CategoryTheory.CategoryStruct.id W.Q ◫ τ₁ = CategoryTheory.CategoryStruct.id W.Q ◫ τ₂)
Goal: τ₁ = τ₂
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.functor_obj_obj_obj
@[simps!]
def functor : (W.Localization ⥤ D) ⥤ W.FunctorsInverting D 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D] (X : CategoryTheory.Functor W.Localization D)
Goal: ∀ (X_1 : C),
  ((CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.functor W D).obj X).obj.obj X_1 =
    X.obj (W.Q.obj X_1)
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.functor_obj_obj_map
@[simps!]
def functor : (W.Localization ⥤ D) ⥤ W.FunctorsInverting D 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D] (X : CategoryTheory.Functor W.Localization D)
Goal: ∀ {X_1 Y : C} (f : X_1 ⟶ Y),
  ((CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.functor W D).obj X).obj.map f = X.map (W.Q.map f)
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.functor_map_app
@[simps!]
def functor : (W.Localization ⥤ D) ⥤ W.FunctorsInverting D 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D]
Goal: ∀ {X Y : CategoryTheory.Functor W.Localization D} (f : X ⟶ Y) (X_1 : C),
  ((CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.functor W D).map f).app X_1 = f.app (W.Q.obj X_1)
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.inverse_obj_obj
@[simps!]
def inverse : W.FunctorsInverting D ⥤ W.Localization ⥤ D 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D] (G : W.FunctorsInverting D) (a : CategoryTheory.Quotient (CategoryTheory.Localization.Construction.relations W))
Goal: ((CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.inverse W D).obj G).obj a = G.obj.obj a.as.obj
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.inverse_map_app
@[simps!]
def inverse : W.FunctorsInverting D ⥤ W.Localization ⥤ D 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D]
Goal: ∀ {X Y : W.FunctorsInverting D} (τ : X ⟶ Y) (X_1 : W.Localization),
  ((CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.inverse W D).map τ).app X_1 =
    CategoryTheory.Localization.Construction.NatTransExtension.app
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ⋯)
        (CategoryTheory.CategoryStruct.comp τ (CategoryTheory.eqToHom ⋯)))
      X_1
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.inverse_obj_map
@[simps!]
def inverse : W.FunctorsInverting D ⥤ W.Localization ⥤ D 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D] (G : W.FunctorsInverting D) (a b : CategoryTheory.Quotient (CategoryTheory.Localization.Construction.relations W)) (hf : a ⟶ b)
Goal: ((CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.inverse W D).obj G).map hf =
  Quot.liftOn hf
    (fun f =>
      CategoryTheory.composePath
        ({ obj := fun X => G.obj.obj X.obj,
              map := fun {X Y} a =>
                Sum.rec (fun val => G.obj.map val) (fun val => CategoryTheory.inv (G.obj.map ↑val)) a }.mapPath
          f))
    ⋯
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.unitIso_inv
@[simps!]
def unitIso : 𝟭 (W.Localization ⥤ D) ≅ functor W D ⋙ inverse W D 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D]
Goal: (CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.unitIso W D).inv = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.unitIso_hom
@[simps!]
def unitIso : 𝟭 (W.Localization ⥤ D) ≅ functor W D ⋙ inverse W D 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D]
Goal: (CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.unitIso W D).hom = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.counitIso_hom
@[simps!]
def counitIso : inverse W D ⋙ functor W D ≅ 𝟭 (W.FunctorsInverting D) 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D]
Goal: (CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.counitIso W D).hom = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.counitIso_inv
@[simps!]
def counitIso : inverse W D ⋙ functor W D ≅ 𝟭 (W.FunctorsInverting D) 
Arguments: {C : Type uC} [CategoryTheory.Category.{uC', uC} C] (W : CategoryTheory.MorphismProperty C) (D : Type uD) [CategoryTheory.Category.{uD', uD} D]
Goal: (CategoryTheory.Localization.Construction.WhiskeringLeftEquivalence.counitIso W D).inv = CategoryTheory.eqToHom ⋯
-- CategoryTheory.pullbackShiftFunctorZero_inv_app
lemma pullbackShiftFunctorZero_inv_app :
    (shiftFunctorZero _ A).inv.app X =
      (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 (by simp)).inv.app X 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] (φ : A →+ B) [CategoryTheory.HasShift C B] (X : CategoryTheory.PullbackShift C φ)
Goal: (CategoryTheory.shiftFunctorZero (CategoryTheory.PullbackShift C φ) A).inv.app X =
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctorZero C B).inv.app X)
    ((CategoryTheory.pullbackShiftIso C φ 0 0 ⋯).inv.app X)
-- CategoryTheory.pullbackShiftFunctorZero_hom_app
lemma pullbackShiftFunctorZero_hom_app :
    (shiftFunctorZero _ A).hom.app X =
      (pullbackShiftIso C φ 0 0 (by simp)).hom.app X ≫ (shiftFunctorZero C B).hom.app X 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] (φ : A →+ B) [CategoryTheory.HasShift C B] (X : CategoryTheory.PullbackShift C φ)
Goal: (CategoryTheory.shiftFunctorZero (CategoryTheory.PullbackShift C φ) A).hom.app X =
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.pullbackShiftIso C φ 0 0 ⋯).hom.app X)
    ((CategoryTheory.shiftFunctorZero C B).hom.app X)
-- CategoryTheory.pullbackShiftFunctorAdd'_inv_app
lemma pullbackShiftFunctorAdd'_inv_app :
    (shiftFunctorAdd' _ a₁ a₂ a₃ h).inv.app X =
      (shiftFunctor (PullbackShift C φ) a₂).map ((pullbackShiftIso C φ a₁ b₁ h₁).hom.app X) ≫
        (pullbackShiftIso C φ a₂ b₂ h₂).hom.app _ ≫
        (shiftFunctorAdd' C b₁ b₂ b₃ (by rw [h₁, h₂, h₃, ← h, φ.map_add])).inv.app X ≫
        (pullbackShiftIso C φ a₃ b₃ h₃).inv.app X 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] (φ : A →+ B) [CategoryTheory.HasShift C B] (X : CategoryTheory.PullbackShift C φ) (a₁ a₂ a₃ : A) (h : a₁ + a₂ = a₃) (b₁ b₂ b₃ : B) (h₁ : b₁ = φ a₁) (h₂ : b₂ = φ a₂) (h₃ : b₃ = φ a₃)
Goal: (CategoryTheory.shiftFunctorAdd' (CategoryTheory.PullbackShift C φ) a₁ a₂ a₃ h).inv.app X =
  CategoryTheory.CategoryStruct.comp
    ((CategoryTheory.shiftFunctor (CategoryTheory.PullbackShift C φ) a₂).map
      ((CategoryTheory.pullbackShiftIso C φ a₁ b₁ h₁).hom.app X))
    (CategoryTheory.CategoryStruct.comp
      ((CategoryTheory.pullbackShiftIso C φ a₂ b₂ h₂).hom.app ((CategoryTheory.shiftFunctor C b₁).obj X))
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctorAdd' C b₁ b₂ b₃ ⋯).inv.app X)
        ((CategoryTheory.pullbackShiftIso C φ a₃ b₃ h₃).inv.app X)))
-- CategoryTheory.pullbackShiftFunctorAdd'_hom_app
lemma pullbackShiftFunctorAdd'_hom_app :
    (shiftFunctorAdd' _ a₁ a₂ a₃ h).hom.app X =
      (pullbackShiftIso C φ a₃ b₃ h₃).hom.app X ≫
      (shiftFunctorAdd' C b₁ b₂ b₃ (by rw [h₁, h₂, h₃, ← h, φ.map_add])).hom.app X ≫
      (pullbackShiftIso C φ a₂ b₂ h₂).inv.app _ ≫
      (shiftFunctor (PullbackShift C φ) a₂).map ((pullbackShiftIso C φ a₁ b₁ h₁).inv.app X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] (φ : A →+ B) [CategoryTheory.HasShift C B] (X : CategoryTheory.PullbackShift C φ) (a₁ a₂ a₃ : A) (h : a₁ + a₂ = a₃) (b₁ b₂ b₃ : B) (h₁ : b₁ = φ a₁) (h₂ : b₂ = φ a₂) (h₃ : b₃ = φ a₃)
Goal: (CategoryTheory.shiftFunctorAdd' (CategoryTheory.PullbackShift C φ) a₁ a₂ a₃ h).hom.app X =
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.pullbackShiftIso C φ a₃ b₃ h₃).hom.app X)
    (CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctorAdd' C b₁ b₂ b₃ ⋯).hom.app X)
      (CategoryTheory.CategoryStruct.comp
        ((CategoryTheory.pullbackShiftIso C φ a₂ b₂ h₂).inv.app ((CategoryTheory.shiftFunctor C b₁).obj X))
        ((CategoryTheory.shiftFunctor (CategoryTheory.PullbackShift C φ) a₂).map
          ((CategoryTheory.pullbackShiftIso C φ a₁ b₁ h₁).inv.app X))))
-- CategoryTheory.Localization.HasSmallLocalizedHom.small
class HasSmallLocalizedHom : Prop where
  small : Small.{w} (W.Q.obj X ⟶ W.Q.obj Y)
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} [self : CategoryTheory.Localization.HasSmallLocalizedHom W X Y]
Goal: Small.{w, max u₁ v₁} (W.Q.obj X ⟶ W.Q.obj Y)
-- CategoryTheory.Localization.hasSmallLocalizedHom_iff
lemma hasSmallLocalizedHom_iff :
    HasSmallLocalizedHom.{w} W X Y ↔ Small.{w} (L.obj X ⟶ L.obj Y) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (W : CategoryTheory.MorphismProperty C) {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (L : CategoryTheory.Functor C D) [L.IsLocalization W] {X Y : C}
Goal: CategoryTheory.Localization.HasSmallLocalizedHom W X Y ↔ Small.{w, v₂} (L.obj X ⟶ L.obj Y)
-- CategoryTheory.Localization.hasSmallLocalizedHom_of_isLocalization
lemma hasSmallLocalizedHom_of_isLocalization :
    HasSmallLocalizedHom.{v₂} W X Y 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (W : CategoryTheory.MorphismProperty C) {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (L : CategoryTheory.Functor C D) [L.IsLocalization W] {X Y : C}
Goal: CategoryTheory.Localization.HasSmallLocalizedHom W X Y
-- CategoryTheory.Localization.small_of_hasSmallLocalizedHom
lemma small_of_hasSmallLocalizedHom [HasSmallLocalizedHom.{w} W X Y] :
    Small.{w} (L.obj X ⟶ L.obj Y) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (W : CategoryTheory.MorphismProperty C) {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (L : CategoryTheory.Functor C D) [L.IsLocalization W] (X Y : C) [CategoryTheory.Localization.HasSmallLocalizedHom W X Y]
Goal: Small.{w, v₂} (L.obj X ⟶ L.obj Y)
-- CategoryTheory.Localization.hasSmallLocalizedHom_iff_of_isos
lemma hasSmallLocalizedHom_iff_of_isos {X' Y' : C} (e : X ≅ X') (e' : Y ≅ Y') :
    HasSmallLocalizedHom.{w} W X Y ↔ HasSmallLocalizedHom.{w} W X' Y' 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (W : CategoryTheory.MorphismProperty C) {X Y X' Y' : C} (e : X ≅ X') (e' : Y ≅ Y')
Goal: CategoryTheory.Localization.HasSmallLocalizedHom W X Y ↔ CategoryTheory.Localization.HasSmallLocalizedHom W X' Y'
-- CategoryTheory.Localization.SmallHom.equiv_equiv_symm
lemma equiv_equiv_symm (L : C ⥤ D) [L.IsLocalization W]
    (L' : C ⥤ D') [L'.IsLocalization W] (G : D ⥤ D')
    (e : L ⋙ G ≅ L') {X Y : C} [HasSmallLocalizedHom.{w} W X Y]
    (f : L.obj X ⟶ L.obj Y) :
    equiv W L' ((equiv W L).symm f) =
      e.inv.app X ≫ G.map f ≫ e.hom.app Y 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (W : CategoryTheory.MorphismProperty C) {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {D' : Type u₃} [CategoryTheory.Category.{v₃, u₃} D'] (L : CategoryTheory.Functor C D) [L.IsLocalization W] (L' : CategoryTheory.Functor C D') [L'.IsLocalization W] (G : CategoryTheory.Functor D D') (e : L.comp G ≅ L') {X Y : C} [CategoryTheory.Localization.HasSmallLocalizedHom W X Y] (f : L.obj X ⟶ L.obj Y)
Goal: (CategoryTheory.Localization.SmallHom.equiv W L') ((CategoryTheory.Localization.SmallHom.equiv W L).symm f) =
  CategoryTheory.CategoryStruct.comp (e.inv.app X) (CategoryTheory.CategoryStruct.comp (G.map f) (e.hom.app Y))
-- CategoryTheory.Localization.SmallHom.equiv_mk
@[simp]
lemma equiv_mk (L : C ⥤ D) [L.IsLocalization W] {X Y : C}
    [HasSmallLocalizedHom.{w} W X Y] (f : X ⟶ Y) :
    equiv.{w} W L (mk W f) = L.map f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (W : CategoryTheory.MorphismProperty C) {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (L : CategoryTheory.Functor C D) [L.IsLocalization W] {X Y : C} [CategoryTheory.Localization.HasSmallLocalizedHom W X Y] (f : X ⟶ Y)
Goal: (CategoryTheory.Localization.SmallHom.equiv W L) (CategoryTheory.Localization.SmallHom.mk W f) = L.map f
-- CategoryTheory.Localization.SmallHom.equiv_mkInv
@[simp]
lemma equiv_mkInv (L : C ⥤ D) [L.IsLocalization W] {X Y : C} (f : Y ⟶ X) (hf : W f)
    [HasSmallLocalizedHom.{w} W X Y] :
    equiv.{w} W L (mkInv f hf) = (Localization.isoOfHom L W f hf).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (L : CategoryTheory.Functor C D) [L.IsLocalization W] {X Y : C} (f : Y ⟶ X) (hf : W f) [CategoryTheory.Localization.HasSmallLocalizedHom W X Y]
Goal: (CategoryTheory.Localization.SmallHom.equiv W L) (CategoryTheory.Localization.SmallHom.mkInv f hf) =
  (CategoryTheory.Localization.isoOfHom L W f hf).inv
-- CategoryTheory.Localization.SmallHom.equiv_comp
lemma equiv_comp (L : C ⥤ D) [L.IsLocalization W] {X Y Z : C} [HasSmallLocalizedHom.{w} W X Y]
    [HasSmallLocalizedHom.{w} W Y Z] [HasSmallLocalizedHom.{w} W X Z]
    (α : SmallHom.{w} W X Y) (β : SmallHom.{w} W Y Z) :
    equiv W L (α.comp β) = equiv W L α ≫ equiv W L β 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (L : CategoryTheory.Functor C D) [L.IsLocalization W] {X Y Z : C} [CategoryTheory.Localization.HasSmallLocalizedHom W X Y] [CategoryTheory.Localization.HasSmallLocalizedHom W Y Z] [CategoryTheory.Localization.HasSmallLocalizedHom W X Z] (α : CategoryTheory.Localization.SmallHom W X Y) (β : CategoryTheory.Localization.SmallHom W Y Z)
Goal: (CategoryTheory.Localization.SmallHom.equiv W L) (α.comp β) =
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.Localization.SmallHom.equiv W L) α)
    ((CategoryTheory.Localization.SmallHom.equiv W L) β)
-- CategoryTheory.Localization.SmallHom.mk_comp_mk
lemma mk_comp_mk [HasSmallLocalizedHom.{w} W X Y] [HasSmallLocalizedHom.{w} W Y Z]
    [HasSmallLocalizedHom.{w} W X Z] (f : X ⟶ Y) (g : Y ⟶ Z) :
    (mk W f).comp (mk W g) = mk W (f ≫ g) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {X Y Z : C} [CategoryTheory.Localization.HasSmallLocalizedHom W X Y] [CategoryTheory.Localization.HasSmallLocalizedHom W Y Z] [CategoryTheory.Localization.HasSmallLocalizedHom W X Z] (f : X ⟶ Y) (g : Y ⟶ Z)
Goal: (CategoryTheory.Localization.SmallHom.mk W f).comp (CategoryTheory.Localization.SmallHom.mk W g) =
  CategoryTheory.Localization.SmallHom.mk W (CategoryTheory.CategoryStruct.comp f g)
-- CategoryTheory.Localization.SmallHom.comp_mk_id
@[simp]
lemma comp_mk_id [HasSmallLocalizedHom.{w} W X Y] [HasSmallLocalizedHom.{w} W Y Y]
    (α : SmallHom.{w} W X Y)  :
    α.comp (mk W (𝟙 Y)) = α 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} [CategoryTheory.Localization.HasSmallLocalizedHom W X Y] [CategoryTheory.Localization.HasSmallLocalizedHom W Y Y] (α : CategoryTheory.Localization.SmallHom W X Y)
Goal: α.comp (CategoryTheory.Localization.SmallHom.mk W (CategoryTheory.CategoryStruct.id Y)) = α
-- CategoryTheory.Localization.SmallHom.mk_id_comp
@[simp]
lemma mk_id_comp [HasSmallLocalizedHom.{w} W X Y] [HasSmallLocalizedHom.{w} W X X]
    (α : SmallHom.{w} W X Y) :
    (mk W (𝟙 X)).comp α = α 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} [CategoryTheory.Localization.HasSmallLocalizedHom W X Y] [CategoryTheory.Localization.HasSmallLocalizedHom W X X] (α : CategoryTheory.Localization.SmallHom W X Y)
Goal: (CategoryTheory.Localization.SmallHom.mk W (CategoryTheory.CategoryStruct.id X)).comp α = α
-- CategoryTheory.Localization.SmallHom.comp_assoc
@[simp]
lemma comp_assoc [HasSmallLocalizedHom.{w} W X Y] [HasSmallLocalizedHom.{w} W X Z]
    [HasSmallLocalizedHom.{w} W X T] [HasSmallLocalizedHom.{w} W Y Z]
    [HasSmallLocalizedHom.{w} W Y T] [HasSmallLocalizedHom.{w} W Z T]
    (α : SmallHom.{w} W X Y) (β : SmallHom.{w} W Y Z) (γ : SmallHom.{w} W Z T) :
    (α.comp β).comp γ = α.comp (β.comp γ) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {X Y Z T : C} [CategoryTheory.Localization.HasSmallLocalizedHom W X Y] [CategoryTheory.Localization.HasSmallLocalizedHom W X Z] [CategoryTheory.Localization.HasSmallLocalizedHom W X T] [CategoryTheory.Localization.HasSmallLocalizedHom W Y Z] [CategoryTheory.Localization.HasSmallLocalizedHom W Y T] [CategoryTheory.Localization.HasSmallLocalizedHom W Z T] (α : CategoryTheory.Localization.SmallHom W X Y) (β : CategoryTheory.Localization.SmallHom W Y Z) (γ : CategoryTheory.Localization.SmallHom W Z T)
Goal: (α.comp β).comp γ = α.comp (β.comp γ)
-- CategoryTheory.Localization.SmallHom.mk_comp_mkInv
@[simp]
lemma mk_comp_mkInv [HasSmallLocalizedHom.{w} W X Y] [HasSmallLocalizedHom.{w} W Y X]
    [HasSmallLocalizedHom.{w} W Y Y] (f : Y ⟶ X) (hf : W f) :
    (mk W f).comp (mkInv f hf) = mk W (𝟙 Y) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} [CategoryTheory.Localization.HasSmallLocalizedHom W X Y] [CategoryTheory.Localization.HasSmallLocalizedHom W Y X] [CategoryTheory.Localization.HasSmallLocalizedHom W Y Y] (f : Y ⟶ X) (hf : W f)
Goal: (CategoryTheory.Localization.SmallHom.mk W f).comp (CategoryTheory.Localization.SmallHom.mkInv f hf) =
  CategoryTheory.Localization.SmallHom.mk W (CategoryTheory.CategoryStruct.id Y)
-- CategoryTheory.Localization.SmallHom.mkInv_comp_mk
@[simp]
lemma mkInv_comp_mk [HasSmallLocalizedHom.{w} W X X] [HasSmallLocalizedHom.{w} W X Y]
    [HasSmallLocalizedHom.{w} W Y X] (f : Y ⟶ X) (hf : W f) :
    (mkInv f hf).comp (mk W f) = mk W (𝟙 X) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} [CategoryTheory.Localization.HasSmallLocalizedHom W X X] [CategoryTheory.Localization.HasSmallLocalizedHom W X Y] [CategoryTheory.Localization.HasSmallLocalizedHom W Y X] (f : Y ⟶ X) (hf : W f)
Goal: (CategoryTheory.Localization.SmallHom.mkInv f hf).comp (CategoryTheory.Localization.SmallHom.mk W f) =
  CategoryTheory.Localization.SmallHom.mk W (CategoryTheory.CategoryStruct.id X)
-- CategoryTheory.LocalizerMorphism.equiv_smallHomMap
lemma equiv_smallHomMap (G : D₁ ⥤ D₂) (e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G)
    (f : SmallHom.{w} W₁ X Y) :
    (SmallHom.equiv W₂ L₂) (Φ.smallHomMap f) =
      e.hom.app X ≫ G.map (SmallHom.equiv W₁ L₁ f) ≫ e.inv.app Y 
Arguments: {C₁ : Type u₁} [CategoryTheory.Category.{v₁, u₁} C₁] {W₁ : CategoryTheory.MorphismProperty C₁} {C₂ : Type u₂} [CategoryTheory.Category.{v₂, u₂} C₂] {W₂ : CategoryTheory.MorphismProperty C₂} {D₁ : Type u₃} [CategoryTheory.Category.{v₃, u₃} D₁] {D₂ : Type u₄} [CategoryTheory.Category.{v₄, u₄} D₂] (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] {X Y : C₁} [CategoryTheory.Localization.HasSmallLocalizedHom W₁ X Y] [CategoryTheory.Localization.HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)] (G : CategoryTheory.Functor D₁ D₂) (e : Φ.functor.comp L₂ ≅ L₁.comp G) (f : CategoryTheory.Localization.SmallHom W₁ X Y)
Goal: (CategoryTheory.Localization.SmallHom.equiv W₂ L₂) (Φ.smallHomMap f) =
  CategoryTheory.CategoryStruct.comp (e.hom.app X)
    (CategoryTheory.CategoryStruct.comp (G.map ((CategoryTheory.Localization.SmallHom.equiv W₁ L₁) f)) (e.inv.app Y))