ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Cokleisli.Adjunction.fromCokleisli_map
@[simps]
def fromCokleisli : Cokleisli U ⥤ C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (U : CategoryTheory.Comonad C) {X : CategoryTheory.Cokleisli U}
Goal: ∀ {x : CategoryTheory.Cokleisli U} (f : X ⟶ x),
  (CategoryTheory.Cokleisli.Adjunction.fromCokleisli U).map f = CategoryTheory.CategoryStruct.comp (U.δ.app X) (U.map f)
-- CategoryTheory.Cokleisli.Adjunction.fromCokleisli_obj
@[simps]
def fromCokleisli : Cokleisli U ⥤ C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (U : CategoryTheory.Comonad C) (X : CategoryTheory.Cokleisli U)
Goal: (CategoryTheory.Cokleisli.Adjunction.fromCokleisli U).obj X = U.obj X
-- CategoryTheory.Comonad.CofreeEqualizer.topMap_f
@[simps!]
def CofreeEqualizer.topMap :  (Comonad.cofree T).obj X.A ⟶ (Comonad.cofree T).obj (T.obj X.A) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Comonad C} (X : T.Coalgebra)
Goal: (CategoryTheory.Comonad.CofreeEqualizer.topMap X).f = T.map X.a
-- CategoryTheory.Comonad.CofreeEqualizer.bottomMap_f
@[simps]
def CofreeEqualizer.bottomMap :
    (Comonad.cofree T).obj X.A ⟶ (Comonad.cofree T).obj (T.obj X.A) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Comonad C} (X : T.Coalgebra)
Goal: (CategoryTheory.Comonad.CofreeEqualizer.bottomMap X).f = T.δ.app X.A
-- CategoryTheory.Comonad.CofreeEqualizer.ι_f
@[simps]
def CofreeEqualizer.ι : X ⟶ (Comonad.cofree T).obj X.A 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Comonad C} (X : T.Coalgebra)
Goal: (CategoryTheory.Comonad.CofreeEqualizer.ι X).f = X.a
-- CategoryTheory.Comonad.CofreeEqualizer.condition
theorem CofreeEqualizer.condition :
    CofreeEqualizer.ι X ≫ CofreeEqualizer.topMap X =
      CofreeEqualizer.ι X ≫ CofreeEqualizer.bottomMap X 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Comonad C} (X : T.Coalgebra)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Comonad.CofreeEqualizer.ι X)
    (CategoryTheory.Comonad.CofreeEqualizer.topMap X) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Comonad.CofreeEqualizer.ι X)
    (CategoryTheory.Comonad.CofreeEqualizer.bottomMap X)
-- CategoryTheory.Comonad.beckCoalgebraFork_π_app
@[simps!]
def beckCoalgebraFork : Fork (CofreeEqualizer.topMap X) (CofreeEqualizer.bottomMap X) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Comonad C} (X : T.Coalgebra)
Goal: ∀ (X_1 : CategoryTheory.Limits.WalkingParallelPair),
  (CategoryTheory.Comonad.beckCoalgebraFork X).π.app X_1 =
    CategoryTheory.Limits.WalkingParallelPair.rec (motive := fun t =>
      X_1 = t →
        (X ⟶
          (CategoryTheory.Limits.parallelPair (CategoryTheory.Comonad.CofreeEqualizer.topMap X)
                (CategoryTheory.Comonad.CofreeEqualizer.bottomMap X)).obj
            X_1))
      (fun h => ⋯ ▸ CategoryTheory.Comonad.CofreeEqualizer.ι X)
      (fun h =>
        ⋯ ▸
          CategoryTheory.CategoryStruct.comp (CategoryTheory.Comonad.CofreeEqualizer.ι X)
            (CategoryTheory.Comonad.CofreeEqualizer.topMap X))
      X_1 ⋯
-- CategoryTheory.Comonad.beckCoalgebraFork_pt
@[simps!]
def beckCoalgebraFork : Fork (CofreeEqualizer.topMap X) (CofreeEqualizer.bottomMap X) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Comonad C} (X : T.Coalgebra)
Goal: (CategoryTheory.Comonad.beckCoalgebraFork X).pt = X
-- CategoryTheory.Comonad.beckFork_pt
@[simps! pt]
def beckFork : Fork (T.map X.a) (T.δ.app _) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Comonad C} (X : T.Coalgebra)
Goal: (CategoryTheory.Comonad.beckFork X).pt = X.A
-- CategoryTheory.Comonad.beckFork_ι
@[simp]
theorem beckFork_ι : (beckFork X).ι = X.a 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Comonad C} (X : T.Coalgebra)
Goal: (CategoryTheory.Comonad.beckFork X).ι = X.a
-- CategoryTheory.Comonad.beckEqualizer_lift
@[simp]
theorem beckEqualizer_lift (s : Fork (T.toFunctor.map X.a) (T.δ.app X.A)) :
    (beckEqualizer X).lift s = s.ι ≫ T.ε.app _ 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Comonad C} (X : T.Coalgebra) (s : CategoryTheory.Limits.Fork (T.map X.a) (T.δ.app X.A))
Goal: (CategoryTheory.Comonad.beckEqualizer X).lift s = CategoryTheory.CategoryStruct.comp s.ι (T.ε.app X.A)
-- CategoryTheory.Dial.tensorObj_src
@[simps] def tensorObj (X Y : Dial C) : Dial C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y : CategoryTheory.Dial C)
Goal: (X.tensorObj Y).src = (X.src ⨯ Y.src)
-- CategoryTheory.Dial.tensorObj_tgt
@[simps] def tensorObj (X Y : Dial C) : Dial C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y : CategoryTheory.Dial C)
Goal: (X.tensorObj Y).tgt = (X.tgt ⨯ Y.tgt)
-- CategoryTheory.Dial.tensorObj_rel
@[simps] def tensorObj (X Y : Dial C) : Dial C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y : CategoryTheory.Dial C)
Goal: (X.tensorObj Y).rel =
  (CategoryTheory.Subobject.pullback
          (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.fst)).obj
      X.rel ⊓
    (CategoryTheory.Subobject.pullback
          (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd)).obj
      Y.rel
-- CategoryTheory.Dial.tensorHom_F
@[simps] def tensorHom {X₁ X₂ Y₁ Y₂ : Dial C} (f : X₁ ⟶ X₂) (g : Y₁ ⟶ Y₂) :
    tensorObj X₁ Y₁ ⟶ tensorObj X₂ Y₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] {X₁ X₂ Y₁ Y₂ : CategoryTheory.Dial C} (f : X₁ ⟶ X₂) (g : Y₁ ⟶ Y₂)
Goal: (CategoryTheory.Dial.tensorHom f g).F =
  CategoryTheory.Limits.prod.lift
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.fst) f.F)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd) g.F)
-- CategoryTheory.Dial.tensorHom_f
@[simps] def tensorHom {X₁ X₂ Y₁ Y₂ : Dial C} (f : X₁ ⟶ X₂) (g : Y₁ ⟶ Y₂) :
    tensorObj X₁ Y₁ ⟶ tensorObj X₂ Y₂ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] {X₁ X₂ Y₁ Y₂ : CategoryTheory.Dial C} (f : X₁ ⟶ X₂) (g : Y₁ ⟶ Y₂)
Goal: (CategoryTheory.Dial.tensorHom f g).f = CategoryTheory.Limits.prod.map f.f g.f
-- CategoryTheory.Dial.tensorUnit_rel
@[simps] def tensorUnit : Dial C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C]
Goal: CategoryTheory.Dial.tensorUnit.rel = ⊤
-- CategoryTheory.Dial.tensorUnit_src
@[simps] def tensorUnit : Dial C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C]
Goal: CategoryTheory.Dial.tensorUnit.src = ⊤_ C
-- CategoryTheory.Dial.tensorUnit_tgt
@[simps] def tensorUnit : Dial C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C]
Goal: CategoryTheory.Dial.tensorUnit.tgt = ⊤_ C
-- CategoryTheory.Dial.leftUnitor_hom_F
@[simps!] def leftUnitor (X : Dial C) : tensorObj tensorUnit X ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: X.leftUnitor.hom.F =
  CategoryTheory.Limits.prod.lift (CategoryTheory.Limits.terminal.from (((⊤_ C) ⨯ X.src) ⨯ X.tgt))
    CategoryTheory.Limits.prod.snd
-- CategoryTheory.Dial.leftUnitor_inv_F
@[simps!] def leftUnitor (X : Dial C) : tensorObj tensorUnit X ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: X.leftUnitor.inv.F = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd
-- CategoryTheory.Dial.leftUnitor_inv_f
@[simps!] def leftUnitor (X : Dial C) : tensorObj tensorUnit X ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: X.leftUnitor.inv.f =
  CategoryTheory.Limits.prod.lift (CategoryTheory.Limits.terminal.from X.src) (CategoryTheory.CategoryStruct.id X.src)
-- CategoryTheory.Dial.leftUnitor_hom_f
@[simps!] def leftUnitor (X : Dial C) : tensorObj tensorUnit X ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: X.leftUnitor.hom.f = CategoryTheory.Limits.prod.snd
-- CategoryTheory.Dial.rightUnitor_hom_F
@[simps!] def rightUnitor (X : Dial C) : tensorObj X tensorUnit ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: X.rightUnitor.hom.F =
  CategoryTheory.Limits.prod.lift CategoryTheory.Limits.prod.snd
    (CategoryTheory.Limits.terminal.from ((X.src ⨯ ⊤_ C) ⨯ X.tgt))
-- CategoryTheory.Dial.rightUnitor_hom_f
@[simps!] def rightUnitor (X : Dial C) : tensorObj X tensorUnit ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: X.rightUnitor.hom.f = CategoryTheory.Limits.prod.fst
-- CategoryTheory.Dial.rightUnitor_inv_F
@[simps!] def rightUnitor (X : Dial C) : tensorObj X tensorUnit ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: X.rightUnitor.inv.F = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst
-- CategoryTheory.Dial.rightUnitor_inv_f
@[simps!] def rightUnitor (X : Dial C) : tensorObj X tensorUnit ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: X.rightUnitor.inv.f =
  CategoryTheory.Limits.prod.lift (CategoryTheory.CategoryStruct.id X.src) (CategoryTheory.Limits.terminal.from X.src)
-- CategoryTheory.Dial.associator_inv_F
@[simps!]
def associator (X Y Z : Dial C) : tensorObj (tensorObj X Y) Z ≅ tensorObj X (tensorObj Y Z) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y Z : CategoryTheory.Dial C)
Goal: (X.associator Y Z).inv.F =
  CategoryTheory.Limits.prod.lift
    (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.fst))
    (CategoryTheory.Limits.prod.lift
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd
        (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.snd))
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd))
-- CategoryTheory.Dial.associator_hom_F
@[simps!]
def associator (X Y Z : Dial C) : tensorObj (tensorObj X Y) Z ≅ tensorObj X (tensorObj Y Z) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y Z : CategoryTheory.Dial C)
Goal: (X.associator Y Z).hom.F =
  CategoryTheory.Limits.prod.lift
    (CategoryTheory.Limits.prod.lift
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst)
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd
        (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst)))
    (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd))
-- CategoryTheory.Dial.associator_hom_f
@[simps!]
def associator (X Y Z : Dial C) : tensorObj (tensorObj X Y) Z ≅ tensorObj X (tensorObj Y Z) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y Z : CategoryTheory.Dial C)
Goal: (X.associator Y Z).hom.f =
  CategoryTheory.Limits.prod.lift
    (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.fst)
    (CategoryTheory.Limits.prod.lift
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.snd)
      CategoryTheory.Limits.prod.snd)
-- CategoryTheory.Dial.associator_inv_f
@[simps!]
def associator (X Y Z : Dial C) : tensorObj (tensorObj X Y) Z ≅ tensorObj X (tensorObj Y Z) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y Z : CategoryTheory.Dial C)
Goal: (X.associator Y Z).inv.f =
  CategoryTheory.Limits.prod.lift
    (CategoryTheory.Limits.prod.lift CategoryTheory.Limits.prod.fst
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst))
    (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd)
-- CategoryTheory.Dial.instMonoidalCategoryStruct_leftUnitor_hom_F
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).hom.F =
  CategoryTheory.Limits.prod.lift (CategoryTheory.Limits.terminal.from (((⊤_ C) ⨯ X.src) ⨯ X.tgt))
    CategoryTheory.Limits.prod.snd
-- CategoryTheory.Dial.instMonoidalCategoryStruct_leftUnitor_inv_F
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).inv.F =
  CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd
-- CategoryTheory.Dial.instMonoidalCategoryStruct_tensorUnit_tgt
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C]
Goal: (𝟙_ (CategoryTheory.Dial C)).tgt = ⊤_ C
-- CategoryTheory.Dial.instMonoidalCategoryStruct_tensorHom_f
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C]
Goal: ∀ {X₁ Y₁ X₂ Y₂ : CategoryTheory.Dial C} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂),
  (CategoryTheory.MonoidalCategory.tensorHom f g).f = CategoryTheory.Limits.prod.map f.f g.f
-- CategoryTheory.Dial.instMonoidalCategoryStruct_leftUnitor_inv_f
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).inv.f =
  CategoryTheory.Limits.prod.lift (CategoryTheory.Limits.terminal.from X.src) (CategoryTheory.CategoryStruct.id X.src)
-- CategoryTheory.Dial.instMonoidalCategoryStruct_rightUnitor_hom_F
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).hom.F =
  CategoryTheory.Limits.prod.lift CategoryTheory.Limits.prod.snd
    (CategoryTheory.Limits.terminal.from ((X.src ⨯ ⊤_ C) ⨯ X.tgt))
-- CategoryTheory.Dial.instMonoidalCategoryStruct_associator_hom_F
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y Z : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.associator X Y Z).hom.F =
  CategoryTheory.Limits.prod.lift
    (CategoryTheory.Limits.prod.lift
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst)
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd
        (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst)))
    (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd))
-- CategoryTheory.Dial.instMonoidalCategoryStruct_tensorObj_tgt
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.tensorObj X Y).tgt = (X.tgt ⨯ Y.tgt)
-- CategoryTheory.Dial.instMonoidalCategoryStruct_associator_hom_f
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y Z : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.associator X Y Z).hom.f =
  CategoryTheory.Limits.prod.lift
    (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.fst)
    (CategoryTheory.Limits.prod.lift
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.snd)
      CategoryTheory.Limits.prod.snd)
-- CategoryTheory.Dial.instMonoidalCategoryStruct_rightUnitor_inv_F
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).inv.F =
  CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst
-- CategoryTheory.Dial.instMonoidalCategoryStruct_tensorObj_src
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.tensorObj X Y).src = (X.src ⨯ Y.src)
-- CategoryTheory.Dial.instMonoidalCategoryStruct_tensorUnit_src
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C]
Goal: (𝟙_ (CategoryTheory.Dial C)).src = ⊤_ C
-- CategoryTheory.Dial.instMonoidalCategoryStruct_whiskerRight_F
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C]
Goal: ∀ {X₁ X₂ : CategoryTheory.Dial C} (f : X₁ ⟶ X₂) (Y : CategoryTheory.Dial C),
  (CategoryTheory.MonoidalCategory.whiskerRight f Y).F =
    CategoryTheory.Limits.prod.lift
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.fst) f.F)
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd)
-- CategoryTheory.Dial.instMonoidalCategoryStruct_whiskerLeft_f
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: ∀ (x x_1 : CategoryTheory.Dial C) (f : x ⟶ x_1),
  (CategoryTheory.MonoidalCategory.whiskerLeft X f).f =
    CategoryTheory.Limits.prod.map (CategoryTheory.CategoryStruct.id X.src) f.f
-- CategoryTheory.Dial.instMonoidalCategoryStruct_associator_inv_f
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y Z : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.associator X Y Z).inv.f =
  CategoryTheory.Limits.prod.lift
    (CategoryTheory.Limits.prod.lift CategoryTheory.Limits.prod.fst
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst))
    (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd)
-- CategoryTheory.Dial.instMonoidalCategoryStruct_whiskerLeft_F
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C)
Goal: ∀ (x x_1 : CategoryTheory.Dial C) (f : x ⟶ x_1),
  (CategoryTheory.MonoidalCategory.whiskerLeft X f).F =
    CategoryTheory.Limits.prod.lift
      (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd) f.F)
-- CategoryTheory.Dial.instMonoidalCategoryStruct_whiskerRight_f
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C]
Goal: ∀ {X₁ X₂ : CategoryTheory.Dial C} (f : X₁ ⟶ X₂) (Y : CategoryTheory.Dial C),
  (CategoryTheory.MonoidalCategory.whiskerRight f Y).f =
    CategoryTheory.Limits.prod.map f.f (CategoryTheory.CategoryStruct.id Y.src)
-- CategoryTheory.Dial.instMonoidalCategoryStruct_tensorUnit_rel
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C]
Goal: (𝟙_ (CategoryTheory.Dial C)).rel = ⊤
-- CategoryTheory.Dial.instMonoidalCategoryStruct_tensorObj_rel
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C] (X Y : CategoryTheory.Dial C)
Goal: (CategoryTheory.MonoidalCategory.tensorObj X Y).rel =
  (CategoryTheory.Subobject.pullback
          (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.fst)).obj
      X.rel ⊓
    (CategoryTheory.Subobject.pullback
          (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd)).obj
      Y.rel
-- CategoryTheory.Dial.instMonoidalCategoryStruct_tensorHom_F
@[simps!]
instance : MonoidalCategoryStruct (Dial C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasPullbacks C]
Goal: ∀ {X₁ Y₁ X₂ Y₂ : CategoryTheory.Dial C} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂),
  (CategoryTheory.MonoidalCategory.tensorHom f g).F =
    CategoryTheory.Limits.prod.lift
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.fst) f.F)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Limits.prod.map CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd) g.F)