l3lab/ntp-mathlib huggingface dataset:
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
⊢ (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 (Functor.Elements F)

Next tactic:
refine' Functor.ext _ _
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · 

Proof state:
case refine'_1
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
⊢ ∀ (X : Functor.Elements F),
    ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj X = (𝟭 (Functor.Elements F)).obj X

Next tactic:
intro X
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    

Proof state:
case refine'_1
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X : Functor.Elements F
⊢ ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj X = (𝟭 (Functor.Elements F)).obj X

Next tactic:
exact Functor.Elements.ext _ _ rfl (by simp [yonedaEquiv])
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    exact Functor.Elements.ext _ _ rfl (by 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X : Functor.Elements F
⊢ F.map
      (eqToHom
        (_ :
          (((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj X).fst =
            (((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj X).fst))
      (((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj X).snd =
    ((𝟭 (Functor.Elements F)).obj X).snd

Next tactic:
simp [yonedaEquiv]
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    exact Functor.Elements.ext _ _ rfl (by simp [yonedaEquiv])
  · 

Proof state:
case refine'_2
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
⊢ ∀ (X Y : Functor.Elements F) (f : X ⟶ Y),
    ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).map f =
      eqToHom (_ : ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj X = (𝟭 (Functor.Elements F)).obj X) ≫
        (𝟭 (Functor.Elements F)).map f ≫
          eqToHom
            (_ : (𝟭 (Functor.Elements F)).obj Y = ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj Y)

Next tactic:
intro X Y f
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    exact Functor.Elements.ext _ _ rfl (by simp [yonedaEquiv])
  · intro X Y f
    

Proof state:
case refine'_2
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X Y : Functor.Elements F
f : X ⟶ Y
⊢ ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).map f =
    eqToHom (_ : ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj X = (𝟭 (Functor.Elements F)).obj X) ≫
      (𝟭 (Functor.Elements F)).map f ≫
        eqToHom (_ : (𝟭 (Functor.Elements F)).obj Y = ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj Y)

Next tactic:
have : ∀ {a b : F.Elements} (H : a = b),
        (eqToHom H).1 = eqToHom (show a.fst = b.fst by cases H; rfl) := by
      rintro _ _ rfl
      simp
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    exact Functor.Elements.ext _ _ rfl (by simp [yonedaEquiv])
  · intro X Y f
    have : ∀ {a b : F.Elements} (H : a = b),
        (eqToHom H).1 = eqToHom (show a.fst = b.fst by 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X Y : Functor.Elements F
f : X ⟶ Y
a b : Functor.Elements F
H : a = b
⊢ a.fst = b.fst

Next tactic:
cases H
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    exact Functor.Elements.ext _ _ rfl (by simp [yonedaEquiv])
  · intro X Y f
    have : ∀ {a b : F.Elements} (H : a = b),
        (eqToHom H).1 = eqToHom (show a.fst = b.fst by cases H; 

Proof state:
case refl
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X Y : Functor.Elements F
f : X ⟶ Y
a : Functor.Elements F
⊢ a.fst = a.fst

Next tactic:
rfl
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    exact Functor.Elements.ext _ _ rfl (by simp [yonedaEquiv])
  · intro X Y f
    have : ∀ {a b : F.Elements} (H : a = b),
        (eqToHom H).1 = eqToHom (show a.fst = b.fst by cases H; rfl) := by
      

Proof state:
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X Y : Functor.Elements F
f : X ⟶ Y
⊢ ∀ {a b : Functor.Elements F} (H : a = b), ↑(eqToHom H) = eqToHom (_ : a.fst = b.fst)

Next tactic:
rintro _ _ rfl
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    exact Functor.Elements.ext _ _ rfl (by simp [yonedaEquiv])
  · intro X Y f
    have : ∀ {a b : F.Elements} (H : a = b),
        (eqToHom H).1 = eqToHom (show a.fst = b.fst by cases H; rfl) := by
      rintro _ _ rfl
      

Proof state:
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X Y : Functor.Elements F
f : X ⟶ Y
a✝ : Functor.Elements F
⊢ ↑(eqToHom (_ : a✝ = a✝)) = eqToHom (_ : a✝.fst = a✝.fst)

Next tactic:
simp
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    exact Functor.Elements.ext _ _ rfl (by simp [yonedaEquiv])
  · intro X Y f
    have : ∀ {a b : F.Elements} (H : a = b),
        (eqToHom H).1 = eqToHom (show a.fst = b.fst by cases H; rfl) := by
      rintro _ _ rfl
      simp
    

Proof state:
case refine'_2
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X Y : Functor.Elements F
f : X ⟶ Y
this : ∀ {a b : Functor.Elements F} (H : a = b), ↑(eqToHom H) = eqToHom (_ : a.fst = b.fst)
⊢ ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).map f =
    eqToHom (_ : ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj X = (𝟭 (Functor.Elements F)).obj X) ≫
      (𝟭 (Functor.Elements F)).map f ≫
        eqToHom (_ : (𝟭 (Functor.Elements F)).obj Y = ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj Y)

Next tactic:
ext
/-- The unit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem from_toCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    exact Functor.Elements.ext _ _ rfl (by simp [yonedaEquiv])
  · intro X Y f
    have : ∀ {a b : F.Elements} (H : a = b),
        (eqToHom H).1 = eqToHom (show a.fst = b.fst by cases H; rfl) := by
      rintro _ _ rfl
      simp
    ext
    

Proof state:
case refine'_2.w
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X Y : Functor.Elements F
f : X ⟶ Y
this : ∀ {a b : Functor.Elements F} (H : a = b), ↑(eqToHom H) = eqToHom (_ : a.fst = b.fst)
⊢ ↑(((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).map f) =
    ↑(eqToHom (_ : ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj X = (𝟭 (Functor.Elements F)).obj X) ≫
        (𝟭 (Functor.Elements F)).map f ≫
          eqToHom
            (_ : (𝟭 (Functor.Elements F)).obj Y = ((toCostructuredArrow F).rightOp ⋙ fromCostructuredArrow F).obj Y))

Next tactic:
simp [this]
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
⊢ (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 (CostructuredArrow yoneda F)

Next tactic:
refine' Functor.ext _ _
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · 

Proof state:
case refine'_1
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
⊢ ∀ (X : CostructuredArrow yoneda F),
    ((fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F).obj X = (𝟭 (CostructuredArrow yoneda F)).obj X

Next tactic:
intro X
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    

Proof state:
case refine'_1
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X : CostructuredArrow yoneda F
⊢ ((fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F).obj X = (𝟭 (CostructuredArrow yoneda F)).obj X

Next tactic:
cases' X with X_left X_right X_hom
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    cases' X with X_left X_right X_hom
    

Proof state:
case refine'_1.mk
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X_left : C
X_right : Discrete PUnit.{1}
X_hom : yoneda.obj X_left ⟶ (Functor.fromPUnit F).obj X_right
⊢ ((fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F).obj { left := X_left, right := X_right, hom := X_hom } =
    (𝟭 (CostructuredArrow yoneda F)).obj { left := X_left, right := X_right, hom := X_hom }

Next tactic:
cases X_right
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    cases' X with X_left X_right X_hom
    cases X_right
    

Proof state:
case refine'_1.mk.mk
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X_left : C
as✝ : PUnit.{1}
X_hom : yoneda.obj X_left ⟶ (Functor.fromPUnit F).obj { as := as✝ }
⊢ ((fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F).obj
      { left := X_left, right := { as := as✝ }, hom := X_hom } =
    (𝟭 (CostructuredArrow yoneda F)).obj { left := X_left, right := { as := as✝ }, hom := X_hom }

Next tactic:
simp only [Functor.id_obj, Functor.rightOp_obj, toCostructuredArrow_obj, Functor.comp_obj,
      CostructuredArrow.mk]
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    cases' X with X_left X_right X_hom
    cases X_right
    simp only [Functor.id_obj, Functor.rightOp_obj, toCostructuredArrow_obj, Functor.comp_obj,
      CostructuredArrow.mk]
    

Proof state:
case refine'_1.mk.mk
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X_left : C
as✝ : PUnit.{1}
X_hom : yoneda.obj X_left ⟶ (Functor.fromPUnit F).obj { as := as✝ }
⊢ {
      left :=
        (op
                ((fromCostructuredArrow F).obj
                  (op { left := X_left, right := { as := as✝ }, hom := X_hom }))).unop.fst.unop,
      right := { as := PUnit.unit },
      hom :=
        (yonedaSections
              (op
                      ((fromCostructuredArrow F).obj
                        (op { left := X_left, right := { as := as✝ }, hom := X_hom }))).unop.fst.unop
              F).inv
          {
            down :=
              (op
                    ((fromCostructuredArrow F).obj
                      (op { left := X_left, right := { as := as✝ }, hom := X_hom }))).unop.snd } } =
    { left := X_left, right := { as := as✝ }, hom := X_hom }

Next tactic:
congr
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    cases' X with X_left X_right X_hom
    cases X_right
    simp only [Functor.id_obj, Functor.rightOp_obj, toCostructuredArrow_obj, Functor.comp_obj,
      CostructuredArrow.mk]
    congr
    

Proof state:
case refine'_1.mk.mk.e_hom
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X_left : C
as✝ : PUnit.{1}
X_hom : yoneda.obj X_left ⟶ (Functor.fromPUnit F).obj { as := as✝ }
⊢ (yonedaSections
          (op
                  ((fromCostructuredArrow F).obj
                    (op { left := X_left, right := { as := as✝ }, hom := X_hom }))).unop.fst.unop
          F).inv
      {
        down :=
          (op
                ((fromCostructuredArrow F).obj
                  (op { left := X_left, right := { as := as✝ }, hom := X_hom }))).unop.snd } =
    X_hom

Next tactic:
ext x f
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    cases' X with X_left X_right X_hom
    cases X_right
    simp only [Functor.id_obj, Functor.rightOp_obj, toCostructuredArrow_obj, Functor.comp_obj,
      CostructuredArrow.mk]
    congr
    ext x f
    

Proof state:
case refine'_1.mk.mk.e_hom.w.h.h
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X_left : C
as✝ : PUnit.{1}
X_hom : yoneda.obj X_left ⟶ (Functor.fromPUnit F).obj { as := as✝ }
x : Cᵒᵖ
f :
  (yoneda.obj
        (op
                ((fromCostructuredArrow F).obj
                  (op { left := X_left, right := { as := as✝ }, hom := X_hom }))).unop.fst.unop).obj
    x
⊢ ((yonedaSections
              (op
                      ((fromCostructuredArrow F).obj
                        (op { left := X_left, right := { as := as✝ }, hom := X_hom }))).unop.fst.unop
              F).inv
          {
            down :=
              (op
                    ((fromCostructuredArrow F).obj
                      (op { left := X_left, right := { as := as✝ }, hom := X_hom }))).unop.snd }).app
      x f =
    X_hom.app x f

Next tactic:
convert congr_fun (X_hom.naturality f.op).symm (𝟙 X_left)
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    cases' X with X_left X_right X_hom
    cases X_right
    simp only [Functor.id_obj, Functor.rightOp_obj, toCostructuredArrow_obj, Functor.comp_obj,
      CostructuredArrow.mk]
    congr
    ext x f
    convert congr_fun (X_hom.naturality f.op).symm (𝟙 X_left)
    

Proof state:
case h.e'_3.h
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
X_left : C
as✝ : PUnit.{1}
X_hom : yoneda.obj X_left ⟶ (Functor.fromPUnit F).obj { as := as✝ }
x : Cᵒᵖ
f :
  (yoneda.obj
        (op
                ((fromCostructuredArrow F).obj
                  (op { left := X_left, right := { as := as✝ }, hom := X_hom }))).unop.fst.unop).obj
    x
e_1✝ : F.obj x = ((Functor.fromPUnit F).obj { as := as✝ }).obj (op x.unop)
⊢ X_hom.app x f = ((yoneda.obj X_left).map f.op ≫ X_hom.app (op x.unop)) (𝟙 X_left)

Next tactic:
simp
/-- The counit of the equivalence `F.Elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. -/
theorem to_fromCostructuredArrow_eq (F : Cᵒᵖ ⥤ Type v) :
    (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 _ := by
  refine' Functor.ext _ _
  · intro X
    cases' X with X_left X_right X_hom
    cases X_right
    simp only [Functor.id_obj, Functor.rightOp_obj, toCostructuredArrow_obj, Functor.comp_obj,
      CostructuredArrow.mk]
    congr
    ext x f
    convert congr_fun (X_hom.naturality f.op).symm (𝟙 X_left)
    simp
  · 

Proof state:
case refine'_2
C : Type u
inst✝ : Category.{v, u} C
F✝ : C ⥤ Type w
F : Cᵒᵖ ⥤ Type v
⊢ ∀ (X Y : CostructuredArrow yoneda F) (f : X ⟶ Y),
    ((fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F).map f =
      eqToHom
          (_ :
            ((fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F).obj X =
              (𝟭 (CostructuredArrow yoneda F)).obj X) ≫
        (𝟭 (CostructuredArrow yoneda F)).map f ≫
          eqToHom
            (_ :
              (𝟭 (CostructuredArrow yoneda F)).obj Y =
                ((fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F).obj Y)

Next tactic:
aesop
/-- The equivalence `(-.Elements)ᵒᵖ ≅ (yoneda, -)` of is actually a natural isomorphism of functors.
-/
theorem costructuredArrow_yoneda_equivalence_naturality {F₁ F₂ : Cᵒᵖ ⥤ Type v} (α : F₁ ⟶ F₂) :
    (map α).op ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
F₁ F₂ : Cᵒᵖ ⥤ Type v
α : F₁ ⟶ F₂
⊢ (map α).op ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α

Next tactic:
fapply Functor.ext
/-- The equivalence `(-.Elements)ᵒᵖ ≅ (yoneda, -)` of is actually a natural isomorphism of functors.
-/
theorem costructuredArrow_yoneda_equivalence_naturality {F₁ F₂ : Cᵒᵖ ⥤ Type v} (α : F₁ ⟶ F₂) :
    (map α).op ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α := by
  fapply Functor.ext
  · 

Proof state:
case h_obj
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
F₁ F₂ : Cᵒᵖ ⥤ Type v
α : F₁ ⟶ F₂
⊢ ∀ (X : (Functor.Elements F₁)ᵒᵖ),
    ((map α).op ⋙ toCostructuredArrow F₂).obj X = (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X

Next tactic:
intro X
/-- The equivalence `(-.Elements)ᵒᵖ ≅ (yoneda, -)` of is actually a natural isomorphism of functors.
-/
theorem costructuredArrow_yoneda_equivalence_naturality {F₁ F₂ : Cᵒᵖ ⥤ Type v} (α : F₁ ⟶ F₂) :
    (map α).op ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α := by
  fapply Functor.ext
  · intro X
    

Proof state:
case h_obj
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
F₁ F₂ : Cᵒᵖ ⥤ Type v
α : F₁ ⟶ F₂
X : (Functor.Elements F₁)ᵒᵖ
⊢ ((map α).op ⋙ toCostructuredArrow F₂).obj X = (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X

Next tactic:
simp only [CostructuredArrow.map_mk, toCostructuredArrow_obj, Functor.op_obj,
      Functor.comp_obj]
/-- The equivalence `(-.Elements)ᵒᵖ ≅ (yoneda, -)` of is actually a natural isomorphism of functors.
-/
theorem costructuredArrow_yoneda_equivalence_naturality {F₁ F₂ : Cᵒᵖ ⥤ Type v} (α : F₁ ⟶ F₂) :
    (map α).op ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α := by
  fapply Functor.ext
  · intro X
    simp only [CostructuredArrow.map_mk, toCostructuredArrow_obj, Functor.op_obj,
      Functor.comp_obj]
    

Proof state:
case h_obj
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
F₁ F₂ : Cᵒᵖ ⥤ Type v
α : F₁ ⟶ F₂
X : (Functor.Elements F₁)ᵒᵖ
⊢ CostructuredArrow.mk
      ((yonedaSections (op ((map α).obj X.unop)).unop.fst.unop F₂).inv { down := (op ((map α).obj X.unop)).unop.snd }) =
    CostructuredArrow.mk ((yonedaSections X.unop.fst.unop F₁).inv { down := X.unop.snd } ≫ α)

Next tactic:
congr
/-- The equivalence `(-.Elements)ᵒᵖ ≅ (yoneda, -)` of is actually a natural isomorphism of functors.
-/
theorem costructuredArrow_yoneda_equivalence_naturality {F₁ F₂ : Cᵒᵖ ⥤ Type v} (α : F₁ ⟶ F₂) :
    (map α).op ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α := by
  fapply Functor.ext
  · intro X
    simp only [CostructuredArrow.map_mk, toCostructuredArrow_obj, Functor.op_obj,
      Functor.comp_obj]
    congr
    

Proof state:
case h_obj.e_f
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
F₁ F₂ : Cᵒᵖ ⥤ Type v
α : F₁ ⟶ F₂
X : (Functor.Elements F₁)ᵒᵖ
⊢ (yonedaSections (op ((map α).obj X.unop)).unop.fst.unop F₂).inv { down := (op ((map α).obj X.unop)).unop.snd } =
    (yonedaSections X.unop.fst.unop F₁).inv { down := X.unop.snd } ≫ α

Next tactic:
ext _ f
/-- The equivalence `(-.Elements)ᵒᵖ ≅ (yoneda, -)` of is actually a natural isomorphism of functors.
-/
theorem costructuredArrow_yoneda_equivalence_naturality {F₁ F₂ : Cᵒᵖ ⥤ Type v} (α : F₁ ⟶ F₂) :
    (map α).op ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α := by
  fapply Functor.ext
  · intro X
    simp only [CostructuredArrow.map_mk, toCostructuredArrow_obj, Functor.op_obj,
      Functor.comp_obj]
    congr
    ext _ f
    

Proof state:
case h_obj.e_f.w.h.h
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
F₁ F₂ : Cᵒᵖ ⥤ Type v
α : F₁ ⟶ F₂
X : (Functor.Elements F₁)ᵒᵖ
x✝ : Cᵒᵖ
f : (yoneda.obj (op ((map α).obj X.unop)).unop.fst.unop).obj x✝
⊢ ((yonedaSections (op ((map α).obj X.unop)).unop.fst.unop F₂).inv { down := (op ((map α).obj X.unop)).unop.snd }).app
      x✝ f =
    ((yonedaSections X.unop.fst.unop F₁).inv { down := X.unop.snd } ≫ α).app x✝ f

Next tactic:
simpa using congr_fun (α.naturality f.op).symm (unop X).snd
/-- The equivalence `(-.Elements)ᵒᵖ ≅ (yoneda, -)` of is actually a natural isomorphism of functors.
-/
theorem costructuredArrow_yoneda_equivalence_naturality {F₁ F₂ : Cᵒᵖ ⥤ Type v} (α : F₁ ⟶ F₂) :
    (map α).op ⋙ toCostructuredArrow F₂ = toCostructuredArrow F₁ ⋙ CostructuredArrow.map α := by
  fapply Functor.ext
  · intro X
    simp only [CostructuredArrow.map_mk, toCostructuredArrow_obj, Functor.op_obj,
      Functor.comp_obj]
    congr
    ext _ f
    simpa using congr_fun (α.naturality f.op).symm (unop X).snd
  · 

Proof state:
case h_map
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
F₁ F₂ : Cᵒᵖ ⥤ Type v
α : F₁ ⟶ F₂
⊢ autoParam
    (∀ (X Y : (Functor.Elements F₁)ᵒᵖ) (f : X ⟶ Y),
      ((map α).op ⋙ toCostructuredArrow F₂).map f =
        eqToHom
            (_ :
              ((map α).op ⋙ toCostructuredArrow F₂).obj X = (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj X) ≫
          (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).map f ≫
            eqToHom
              (_ :
                (toCostructuredArrow F₁ ⋙ CostructuredArrow.map α).obj Y = ((map α).op ⋙ toCostructuredArrow F₂).obj Y))
    _auto✝

Next tactic:
simp [autoParam]