-- Module: Mathlib.CategoryTheory.WithTerminal.Basic
Goal:
C : Type u
inst : Category.{v, u} C
x_1 : (WithTerminal C)ᵒᵖ
x_2 : WithTerminal C
x : C
simp :
  Eq.{v + 1} (α := WithInitial.of (op x) ⟶ WithInitial.of (op x)) (Quiver.Hom.op (down (unop (𝟙 (op (of x))))))
    (𝟙 (WithInitial.of (op x)))
⊢ Eq.{v + 1} (α :=
    Quiver.Hom (V := WithInitial Cᵒᵖ)
      (Functor.obj.{v, v, u, u}
        {
          obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
            match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
            | op x =>
              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
              | of x => WithInitial.of (op x)
              | star => WithInitial.star,
          map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
            match (motive :=
              (x ⟶ y) →
                Quiver.Hom (V := WithInitial Cᵒᵖ)
                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star :
                    WithInitial Cᵒᵖ)
                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star :
                    WithInitial Cᵒᵖ))
              x_3 with
            | op f =>
              match (motive :=
                (x y : (WithTerminal C)ᵒᵖ) →
                  (unop y ⟶ unop x) →
                    (x ⟶ y) →
                      Quiver.Hom (V := WithInitial Cᵒᵖ)
                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                          | op x =>
                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                            | of x => WithInitial.of (op x)
                            | star => WithInitial.star :
                          WithInitial Cᵒᵖ)
                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                          | op x =>
                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                            | of x => WithInitial.of (op x)
                            | star => WithInitial.star :
                          WithInitial Cᵒᵖ))
                x, y, f, x_3 with
              | op (of x), op (of y), f, x_4 => (down f).op
              | op star, op (of a), x, x_4 =>
                Limits.IsInitial.to WithInitial.starInitial
                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                  | op x =>
                    match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                    | of x => WithInitial.of (op x)
                    | star => WithInitial.star)
              | op star, op star, x, x_4 =>
                𝟙
                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                  | op x =>
                    match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                    | of x => WithInitial.of (op x)
                    | star => WithInitial.star),
          map_id := ⋯, map_comp := ⋯ }
        (op (of x)))
      (Functor.obj.{v, v, u, u}
        {
          obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
            match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
            | op x =>
              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
              | of x => WithInitial.of (op x)
              | star => WithInitial.star,
          map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
            match (motive :=
              (x ⟶ y) →
                Quiver.Hom (V := WithInitial Cᵒᵖ)
                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star :
                    WithInitial Cᵒᵖ)
                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star :
                    WithInitial Cᵒᵖ))
              x_3 with
            | op f =>
              match (motive :=
                (x y : (WithTerminal C)ᵒᵖ) →
                  (unop y ⟶ unop x) →
                    (x ⟶ y) →
                      Quiver.Hom (V := WithInitial Cᵒᵖ)
                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                          | op x =>
                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                            | of x => WithInitial.of (op x)
                            | star => WithInitial.star :
                          WithInitial Cᵒᵖ)
                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                          | op x =>
                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                            | of x => WithInitial.of (op x)
                            | star => WithInitial.star :
                          WithInitial Cᵒᵖ))
                x, y, f, x_3 with
              | op (of x), op (of y), f, x_4 => (down f).op
              | op star, op (of a), x, x_4 =>
                Limits.IsInitial.to WithInitial.starInitial
                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                  | op x =>
                    match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                    | of x => WithInitial.of (op x)
                    | star => WithInitial.star)
              | op star, op star, x, x_4 =>
                𝟙
                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                  | op x =>
                    match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                    | of x => WithInitial.of (op x)
                    | star => WithInitial.star),
          map_id := ⋯, map_comp := ⋯ }
        (op (of x))))
    (Functor.map
          {
            obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
              match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
              | op x =>
                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                | of x => WithInitial.of (op x)
                | star => WithInitial.star,
            map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
              match (motive :=
                (x ⟶ y) →
                  Quiver.Hom (V := WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ))
                x_3 with
              | op f =>
                match (motive :=
                  (x y : (WithTerminal C)ᵒᵖ) →
                    (unop y ⟶ unop x) →
                      (x ⟶ y) →
                        Quiver.Hom (V := WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ))
                  x, y, f, x_3 with
                | op (of x), op (of y), f, x_4 => (down f).op
                | op star, op (of a), x, x_4 =>
                  Limits.IsInitial.to WithInitial.starInitial
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star)
                | op star, op star, x, x_4 =>
                  𝟙
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star),
            map_id := ⋯, map_comp := ⋯ }
          (NatTrans.app
            (Iso.hom
              (NatIso.ofComponents
                (fun (x : (WithTerminal C)ᵒᵖ) ↦
                  match (motive :=
                    (x : (WithTerminal C)ᵒᵖ) →
                      Functor.obj.{v, v, u, u} (𝟭 (WithTerminal C)ᵒᵖ) x ≅
                        Functor.obj.{v, v, u, u}
                          (Functor.comp (D := WithInitial Cᵒᵖ)
                            {
                              obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
                                match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                | op x =>
                                  match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                  | of x => WithInitial.of (op x)
                                  | star => WithInitial.star,
                              map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
                                match (motive :=
                                  (x ⟶ y) →
                                    Quiver.Hom (V := WithInitial Cᵒᵖ)
                                      (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                        | op x =>
                                          match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                          | of x => WithInitial.of (op x)
                                          | star => WithInitial.star :
                                        WithInitial Cᵒᵖ)
                                      (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                        | op x =>
                                          match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                          | of x => WithInitial.of (op x)
                                          | star => WithInitial.star :
                                        WithInitial Cᵒᵖ))
                                  x_3 with
                                | op f =>
                                  match (motive :=
                                    (x y : (WithTerminal C)ᵒᵖ) →
                                      (unop y ⟶ unop x) →
                                        (x ⟶ y) →
                                          Quiver.Hom (V := WithInitial Cᵒᵖ)
                                            (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                              | op x =>
                                                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                                | of x => WithInitial.of (op x)
                                                | star => WithInitial.star :
                                              WithInitial Cᵒᵖ)
                                            (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                              | op x =>
                                                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                                | of x => WithInitial.of (op x)
                                                | star => WithInitial.star :
                                              WithInitial Cᵒᵖ))
                                    x, y, f, x_3 with
                                  | op (of x), op (of y), f, x_4 => (down f).op
                                  | op star, op (of a), x, x_4 =>
                                    Limits.IsInitial.to WithInitial.starInitial
                                      (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                                      | op x =>
                                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                        | of x => WithInitial.of (op x)
                                        | star => WithInitial.star)
                                  | op star, op star, x, x_4 =>
                                    𝟙
                                      (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                                      | op x =>
                                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                        | of x => WithInitial.of (op x)
                                        | star => WithInitial.star),
                              map_id := ⋯, map_comp := ⋯ }
                            {
                              obj := fun (x : WithInitial Cᵒᵖ) ↦
                                match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                                | WithInitial.of x => op (of (unop x))
                                | WithInitial.star => op star,
                              map := fun {x y : WithInitial Cᵒᵖ} (f : x ⟶ y) ↦
                                match (motive :=
                                  (x y : WithInitial Cᵒᵖ) →
                                    (x ⟶ y) →
                                      Quiver.Hom (V := (WithTerminal C)ᵒᵖ)
                                        (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                                          | WithInitial.of x => op (of (unop x))
                                          | WithInitial.star => op star :
                                          (WithTerminal C)ᵒᵖ)
                                        (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) y with
                                          | WithInitial.of x => op (of (unop x))
                                          | WithInitial.star => op star :
                                          (WithTerminal C)ᵒᵖ))
                                  x, y, f with
                                | WithInitial.of (op x), WithInitial.of (op y), f => WithInitial.down f
                                | WithInitial.star, WithInitial.of (op unop), x =>
                                  op
                                    (Limits.IsTerminal.from starTerminal
                                      (Opposite.unop
                                        (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ)
                                          WithInitial.of (op unop) with
                                        | WithInitial.of x => op (of (Opposite.unop x))
                                        | WithInitial.star => op star)))
                                | WithInitial.star, WithInitial.star, x =>
                                  𝟙
                                    (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) WithInitial.star with
                                    | WithInitial.of x => op (of (unop x))
                                    | WithInitial.star => op star),
                              map_id := ⋯, map_comp := ⋯ })
                          x)
                    x with
                  | op x =>
                    match (motive :=
                      (x : WithTerminal C) →
                        Functor.obj.{v, v, u, u} (𝟭 (WithTerminal C)ᵒᵖ) (op x) ≅
                          Functor.obj.{v, v, u, u}
                            (Functor.comp (D := WithInitial Cᵒᵖ)
                              {
                                obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
                                  match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                  | op x =>
                                    match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                    | of x => WithInitial.of (op x)
                                    | star => WithInitial.star,
                                map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
                                  match (motive :=
                                    (x ⟶ y) →
                                      Quiver.Hom (V := WithInitial Cᵒᵖ)
                                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                          | op x =>
                                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                            | of x => WithInitial.of (op x)
                                            | star => WithInitial.star :
                                          WithInitial Cᵒᵖ)
                                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                          | op x =>
                                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                            | of x => WithInitial.of (op x)
                                            | star => WithInitial.star :
                                          WithInitial Cᵒᵖ))
                                    x_3 with
                                  | op f =>
                                    match (motive :=
                                      (x y : (WithTerminal C)ᵒᵖ) →
                                        (unop y ⟶ unop x) →
                                          (x ⟶ y) →
                                            Quiver.Hom (V := WithInitial Cᵒᵖ)
                                              (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                                | op x =>
                                                  match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                                  | of x => WithInitial.of (op x)
                                                  | star => WithInitial.star :
                                                WithInitial Cᵒᵖ)
                                              (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                                | op x =>
                                                  match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                                  | of x => WithInitial.of (op x)
                                                  | star => WithInitial.star :
                                                WithInitial Cᵒᵖ))
                                      x, y, f, x_3 with
                                    | op (of x), op (of y), f, x_4 => (down f).op
                                    | op star, op (of a), x, x_4 =>
                                      Limits.IsInitial.to WithInitial.starInitial
                                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                                        | op x =>
                                          match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                          | of x => WithInitial.of (op x)
                                          | star => WithInitial.star)
                                    | op star, op star, x, x_4 =>
                                      𝟙
                                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                                        | op x =>
                                          match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                          | of x => WithInitial.of (op x)
                                          | star => WithInitial.star),
                                map_id := ⋯, map_comp := ⋯ }
                              {
                                obj := fun (x : WithInitial Cᵒᵖ) ↦
                                  match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                                  | WithInitial.of x => op (of (unop x))
                                  | WithInitial.star => op star,
                                map := fun {x y : WithInitial Cᵒᵖ} (f : x ⟶ y) ↦
                                  match (motive :=
                                    (x y : WithInitial Cᵒᵖ) →
                                      (x ⟶ y) →
                                        Quiver.Hom (V := (WithTerminal C)ᵒᵖ)
                                          (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                                            | WithInitial.of x => op (of (unop x))
                                            | WithInitial.star => op star :
                                            (WithTerminal C)ᵒᵖ)
                                          (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) y with
                                            | WithInitial.of x => op (of (unop x))
                                            | WithInitial.star => op star :
                                            (WithTerminal C)ᵒᵖ))
                                    x, y, f with
                                  | WithInitial.of (op x), WithInitial.of (op y), f => WithInitial.down f
                                  | WithInitial.star, WithInitial.of (op unop), x =>
                                    op
                                      (Limits.IsTerminal.from starTerminal
                                        (Opposite.unop
                                          (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ)
                                            WithInitial.of (op unop) with
                                          | WithInitial.of x => op (of (Opposite.unop x))
                                          | WithInitial.star => op star)))
                                  | WithInitial.star, WithInitial.star, x =>
                                    𝟙
                                      (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) WithInitial.star with
                                      | WithInitial.of x => op (of (unop x))
                                      | WithInitial.star => op star),
                                map_id := ⋯, map_comp := ⋯ })
                            (op x))
                      x with
                    | of x => Iso.refl (Functor.obj.{v, v, u, u} (𝟭 (WithTerminal C)ᵒᵖ) (op (of x)))
                    | star => Iso.refl (Functor.obj.{v, v, u, u} (𝟭 (WithTerminal C)ᵒᵖ) (op star)))
                ⋯))
            (op (of x))) ≫
        NatTrans.app
          (Iso.hom
            (NatIso.ofComponents
              (fun (x : WithInitial Cᵒᵖ) ↦
                match (motive :=
                  (x : WithInitial Cᵒᵖ) →
                    Functor.obj.{v, v, u, u}
                        (Functor.comp (D := (WithTerminal C)ᵒᵖ)
                          {
                            obj := fun (x : WithInitial Cᵒᵖ) ↦
                              match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                              | WithInitial.of x => op (of (unop x))
                              | WithInitial.star => op star,
                            map := fun {x y : WithInitial Cᵒᵖ} (f : x ⟶ y) ↦
                              match (motive :=
                                (x y : WithInitial Cᵒᵖ) →
                                  (x ⟶ y) →
                                    Quiver.Hom (V := (WithTerminal C)ᵒᵖ)
                                      (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                                        | WithInitial.of x => op (of (unop x))
                                        | WithInitial.star => op star :
                                        (WithTerminal C)ᵒᵖ)
                                      (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) y with
                                        | WithInitial.of x => op (of (unop x))
                                        | WithInitial.star => op star :
                                        (WithTerminal C)ᵒᵖ))
                                x, y, f with
                              | WithInitial.of (op x), WithInitial.of (op y), f => WithInitial.down f
                              | WithInitial.star, WithInitial.of (op unop), x =>
                                op
                                  (Limits.IsTerminal.from starTerminal
                                    (Opposite.unop
                                      (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ)
                                        WithInitial.of (op unop) with
                                      | WithInitial.of x => op (of (Opposite.unop x))
                                      | WithInitial.star => op star)))
                              | WithInitial.star, WithInitial.star, x =>
                                𝟙
                                  (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) WithInitial.star with
                                  | WithInitial.of x => op (of (unop x))
                                  | WithInitial.star => op star),
                            map_id := ⋯, map_comp := ⋯ : Functor.{v, v, u, u} (WithInitial Cᵒᵖ) (WithTerminal C)ᵒᵖ }
                          {
                            obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
                              match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                              | op x =>
                                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                | of x => WithInitial.of (op x)
                                | star => WithInitial.star,
                            map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
                              match (motive :=
                                (x ⟶ y) →
                                  Quiver.Hom (V := WithInitial Cᵒᵖ)
                                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                      | op x =>
                                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                        | of x => WithInitial.of (op x)
                                        | star => WithInitial.star :
                                      WithInitial Cᵒᵖ)
                                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                      | op x =>
                                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                        | of x => WithInitial.of (op x)
                                        | star => WithInitial.star :
                                      WithInitial Cᵒᵖ))
                                x_3 with
                              | op f =>
                                match (motive :=
                                  (x y : (WithTerminal C)ᵒᵖ) →
                                    (unop y ⟶ unop x) →
                                      (x ⟶ y) →
                                        Quiver.Hom (V := WithInitial Cᵒᵖ)
                                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                            | op x =>
                                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                              | of x => WithInitial.of (op x)
                                              | star => WithInitial.star :
                                            WithInitial Cᵒᵖ)
                                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                            | op x =>
                                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                              | of x => WithInitial.of (op x)
                                              | star => WithInitial.star :
                                            WithInitial Cᵒᵖ))
                                  x, y, f, x_3 with
                                | op (of x), op (of y), f, x_4 => (down f).op
                                | op star, op (of a), x, x_4 =>
                                  Limits.IsInitial.to WithInitial.starInitial
                                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                                    | op x =>
                                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                      | of x => WithInitial.of (op x)
                                      | star => WithInitial.star)
                                | op star, op star, x, x_4 =>
                                  𝟙
                                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                                    | op x =>
                                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                      | of x => WithInitial.of (op x)
                                      | star => WithInitial.star),
                            map_id := ⋯, map_comp := ⋯ : Functor.{v, v, u, u} (WithTerminal C)ᵒᵖ (WithInitial Cᵒᵖ) })
                        x ≅
                      Functor.obj.{v, v, u, u} (𝟭 (WithInitial Cᵒᵖ)) x)
                  x with
                | WithInitial.of x =>
                  Iso.refl
                    (Functor.obj.{v, v, u, u}
                      (Functor.comp (D := (WithTerminal C)ᵒᵖ)
                        {
                          obj := fun (x : WithInitial Cᵒᵖ) ↦
                            match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                            | WithInitial.of x => op (of (unop x))
                            | WithInitial.star => op star,
                          map := fun {x y : WithInitial Cᵒᵖ} (f : x ⟶ y) ↦
                            match (motive :=
                              (x y : WithInitial Cᵒᵖ) →
                                (x ⟶ y) →
                                  Quiver.Hom (V := (WithTerminal C)ᵒᵖ)
                                    (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                                      | WithInitial.of x => op (of (unop x))
                                      | WithInitial.star => op star :
                                      (WithTerminal C)ᵒᵖ)
                                    (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) y with
                                      | WithInitial.of x => op (of (unop x))
                                      | WithInitial.star => op star :
                                      (WithTerminal C)ᵒᵖ))
                              x, y, f with
                            | WithInitial.of (op x), WithInitial.of (op y), f => WithInitial.down f
                            | WithInitial.star, WithInitial.of (op unop), x =>
                              op
                                (Limits.IsTerminal.from starTerminal
                                  (Opposite.unop
                                    (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ)
                                      WithInitial.of (op unop) with
                                    | WithInitial.of x => op (of (Opposite.unop x))
                                    | WithInitial.star => op star)))
                            | WithInitial.star, WithInitial.star, x =>
                              𝟙
                                (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) WithInitial.star with
                                | WithInitial.of x => op (of (unop x))
                                | WithInitial.star => op star),
                          map_id := ⋯, map_comp := ⋯ : Functor.{v, v, u, u} (WithInitial Cᵒᵖ) (WithTerminal C)ᵒᵖ }
                        {
                          obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
                            match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star,
                          map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
                            match (motive :=
                              (x ⟶ y) →
                                Quiver.Hom (V := WithInitial Cᵒᵖ)
                                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                    | op x =>
                                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                      | of x => WithInitial.of (op x)
                                      | star => WithInitial.star :
                                    WithInitial Cᵒᵖ)
                                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                    | op x =>
                                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                      | of x => WithInitial.of (op x)
                                      | star => WithInitial.star :
                                    WithInitial Cᵒᵖ))
                              x_3 with
                            | op f =>
                              match (motive :=
                                (x y : (WithTerminal C)ᵒᵖ) →
                                  (unop y ⟶ unop x) →
                                    (x ⟶ y) →
                                      Quiver.Hom (V := WithInitial Cᵒᵖ)
                                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                          | op x =>
                                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                            | of x => WithInitial.of (op x)
                                            | star => WithInitial.star :
                                          WithInitial Cᵒᵖ)
                                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                          | op x =>
                                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                            | of x => WithInitial.of (op x)
                                            | star => WithInitial.star :
                                          WithInitial Cᵒᵖ))
                                x, y, f, x_3 with
                              | op (of x), op (of y), f, x_4 => (down f).op
                              | op star, op (of a), x, x_4 =>
                                Limits.IsInitial.to WithInitial.starInitial
                                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                                  | op x =>
                                    match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                    | of x => WithInitial.of (op x)
                                    | star => WithInitial.star)
                              | op star, op star, x, x_4 =>
                                𝟙
                                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                                  | op x =>
                                    match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                    | of x => WithInitial.of (op x)
                                    | star => WithInitial.star),
                          map_id := ⋯, map_comp := ⋯ : Functor.{v, v, u, u} (WithTerminal C)ᵒᵖ (WithInitial Cᵒᵖ) })
                      (WithInitial.of x))
                | WithInitial.star =>
                  Iso.refl
                    (Functor.obj.{v, v, u, u} (C := WithInitial Cᵒᵖ)
                      (Functor.comp (D := (WithTerminal C)ᵒᵖ)
                        {
                          obj := fun (x : WithInitial Cᵒᵖ) ↦
                            match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                            | WithInitial.of x => op (of (unop x))
                            | WithInitial.star => op star,
                          map := fun {x y : WithInitial Cᵒᵖ} (f : x ⟶ y) ↦
                            match (motive :=
                              (x y : WithInitial Cᵒᵖ) →
                                (x ⟶ y) →
                                  Quiver.Hom (V := (WithTerminal C)ᵒᵖ)
                                    (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) x with
                                      | WithInitial.of x => op (of (unop x))
                                      | WithInitial.star => op star :
                                      (WithTerminal C)ᵒᵖ)
                                    (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) y with
                                      | WithInitial.of x => op (of (unop x))
                                      | WithInitial.star => op star :
                                      (WithTerminal C)ᵒᵖ))
                              x, y, f with
                            | WithInitial.of (op x), WithInitial.of (op y), f => WithInitial.down f
                            | WithInitial.star, WithInitial.of (op unop), x =>
                              op
                                (Limits.IsTerminal.from starTerminal
                                  (Opposite.unop
                                    (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ)
                                      WithInitial.of (op unop) with
                                    | WithInitial.of x => op (of (Opposite.unop x))
                                    | WithInitial.star => op star)))
                            | WithInitial.star, WithInitial.star, x =>
                              𝟙
                                (match (motive := WithInitial Cᵒᵖ → (WithTerminal C)ᵒᵖ) WithInitial.star with
                                | WithInitial.of x => op (of (unop x))
                                | WithInitial.star => op star),
                          map_id := ⋯, map_comp := ⋯ : Functor.{v, v, u, u} (WithInitial Cᵒᵖ) (WithTerminal C)ᵒᵖ }
                        {
                          obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
                            match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star,
                          map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
                            match (motive :=
                              (x ⟶ y) →
                                Quiver.Hom (V := WithInitial Cᵒᵖ)
                                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                    | op x =>
                                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                      | of x => WithInitial.of (op x)
                                      | star => WithInitial.star :
                                    WithInitial Cᵒᵖ)
                                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                    | op x =>
                                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                      | of x => WithInitial.of (op x)
                                      | star => WithInitial.star :
                                    WithInitial Cᵒᵖ))
                              x_3 with
                            | op f =>
                              match (motive :=
                                (x y : (WithTerminal C)ᵒᵖ) →
                                  (unop y ⟶ unop x) →
                                    (x ⟶ y) →
                                      Quiver.Hom (V := WithInitial Cᵒᵖ)
                                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                                          | op x =>
                                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                            | of x => WithInitial.of (op x)
                                            | star => WithInitial.star :
                                          WithInitial Cᵒᵖ)
                                        (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                                          | op x =>
                                            match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                            | of x => WithInitial.of (op x)
                                            | star => WithInitial.star :
                                          WithInitial Cᵒᵖ))
                                x, y, f, x_3 with
                              | op (of x), op (of y), f, x_4 => (down f).op
                              | op star, op (of a), x, x_4 =>
                                Limits.IsInitial.to WithInitial.starInitial
                                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                                  | op x =>
                                    match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                    | of x => WithInitial.of (op x)
                                    | star => WithInitial.star)
                              | op star, op star, x, x_4 =>
                                𝟙
                                  (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                                  | op x =>
                                    match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                    | of x => WithInitial.of (op x)
                                    | star => WithInitial.star),
                          map_id := ⋯, map_comp := ⋯ : Functor.{v, v, u, u} (WithTerminal C)ᵒᵖ (WithInitial Cᵒᵖ) })
                      WithInitial.star))
              ⋯))
          (Functor.obj.{v, v, u, u}
            {
              obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
                match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                | op x =>
                  match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                  | of x => WithInitial.of (op x)
                  | star => WithInitial.star,
              map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
                match (motive :=
                  (x ⟶ y) →
                    Quiver.Hom (V := WithInitial Cᵒᵖ)
                      (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                        | op x =>
                          match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                          | of x => WithInitial.of (op x)
                          | star => WithInitial.star :
                        WithInitial Cᵒᵖ)
                      (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                        | op x =>
                          match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                          | of x => WithInitial.of (op x)
                          | star => WithInitial.star :
                        WithInitial Cᵒᵖ))
                  x_3 with
                | op f =>
                  match (motive :=
                    (x y : (WithTerminal C)ᵒᵖ) →
                      (unop y ⟶ unop x) →
                        (x ⟶ y) →
                          Quiver.Hom (V := WithInitial Cᵒᵖ)
                            (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                              | op x =>
                                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                | of x => WithInitial.of (op x)
                                | star => WithInitial.star :
                              WithInitial Cᵒᵖ)
                            (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                              | op x =>
                                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                                | of x => WithInitial.of (op x)
                                | star => WithInitial.star :
                              WithInitial Cᵒᵖ))
                    x, y, f, x_3 with
                  | op (of x), op (of y), f, x_4 => (down f).op
                  | op star, op (of a), x, x_4 =>
                    Limits.IsInitial.to WithInitial.starInitial
                      (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star)
                  | op star, op star, x, x_4 =>
                    𝟙
                      (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star),
              map_id := ⋯, map_comp := ⋯ : Functor.{v, v, u, u} (WithTerminal C)ᵒᵖ (WithInitial Cᵒᵖ) }
            (op (of x))) :
      Quiver.Hom (V := WithInitial Cᵒᵖ)
        (Functor.obj.{v, v, u, u}
          {
            obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
              match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
              | op x =>
                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                | of x => WithInitial.of (op x)
                | star => WithInitial.star,
            map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
              match (motive :=
                (x ⟶ y) →
                  Quiver.Hom (V := WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ))
                x_3 with
              | op f =>
                match (motive :=
                  (x y : (WithTerminal C)ᵒᵖ) →
                    (unop y ⟶ unop x) →
                      (x ⟶ y) →
                        Quiver.Hom (V := WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ))
                  x, y, f, x_3 with
                | op (of x), op (of y), f, x_4 => (down f).op
                | op star, op (of a), x, x_4 =>
                  Limits.IsInitial.to WithInitial.starInitial
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star)
                | op star, op star, x, x_4 =>
                  𝟙
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star),
            map_id := ⋯, map_comp := ⋯ }
          (op (of x)))
        (Functor.obj.{v, v, u, u}
          {
            obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
              match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
              | op x =>
                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                | of x => WithInitial.of (op x)
                | star => WithInitial.star,
            map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
              match (motive :=
                (x ⟶ y) →
                  Quiver.Hom (V := WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ))
                x_3 with
              | op f =>
                match (motive :=
                  (x y : (WithTerminal C)ᵒᵖ) →
                    (unop y ⟶ unop x) →
                      (x ⟶ y) →
                        Quiver.Hom (V := WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ))
                  x, y, f, x_3 with
                | op (of x), op (of y), f, x_4 => (down f).op
                | op star, op (of a), x, x_4 =>
                  Limits.IsInitial.to WithInitial.starInitial
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star)
                | op star, op star, x, x_4 =>
                  𝟙
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star),
            map_id := ⋯, map_comp := ⋯ }
          (op (of x))))
    (𝟙
        (Functor.obj.{v, v, u, u}
          {
            obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
              match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
              | op x =>
                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                | of x => WithInitial.of (op x)
                | star => WithInitial.star,
            map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
              match (motive :=
                (x ⟶ y) →
                  Quiver.Hom (V := WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ))
                x_3 with
              | op f =>
                match (motive :=
                  (x y : (WithTerminal C)ᵒᵖ) →
                    (unop y ⟶ unop x) →
                      (x ⟶ y) →
                        Quiver.Hom (V := WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ))
                  x, y, f, x_3 with
                | op (of x), op (of y), f, x_4 => (down f).op
                | op star, op (of a), x, x_4 =>
                  Limits.IsInitial.to WithInitial.starInitial
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star)
                | op star, op star, x, x_4 =>
                  𝟙
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star),
            map_id := ⋯, map_comp := ⋯ }
          (op (of x))) :
      Quiver.Hom (V := WithInitial Cᵒᵖ)
        (Functor.obj.{v, v, u, u}
          {
            obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
              match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
              | op x =>
                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                | of x => WithInitial.of (op x)
                | star => WithInitial.star,
            map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
              match (motive :=
                (x ⟶ y) →
                  Quiver.Hom (V := WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ))
                x_3 with
              | op f =>
                match (motive :=
                  (x y : (WithTerminal C)ᵒᵖ) →
                    (unop y ⟶ unop x) →
                      (x ⟶ y) →
                        Quiver.Hom (V := WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ))
                  x, y, f, x_3 with
                | op (of x), op (of y), f, x_4 => (down f).op
                | op star, op (of a), x, x_4 =>
                  Limits.IsInitial.to WithInitial.starInitial
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star)
                | op star, op star, x, x_4 =>
                  𝟙
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star),
            map_id := ⋯, map_comp := ⋯ }
          (op (of x)))
        (Functor.obj.{v, v, u, u}
          {
            obj := fun (x : (WithTerminal C)ᵒᵖ) ↦
              match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
              | op x =>
                match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                | of x => WithInitial.of (op x)
                | star => WithInitial.star,
            map := fun {x y : (WithTerminal C)ᵒᵖ} (x_3 : x ⟶ y) ↦
              match (motive :=
                (x ⟶ y) →
                  Quiver.Hom (V := WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ)
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                      | op x =>
                        match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                        | of x => WithInitial.of (op x)
                        | star => WithInitial.star :
                      WithInitial Cᵒᵖ))
                x_3 with
              | op f =>
                match (motive :=
                  (x y : (WithTerminal C)ᵒᵖ) →
                    (unop y ⟶ unop x) →
                      (x ⟶ y) →
                        Quiver.Hom (V := WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) x with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ)
                          (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) y with
                            | op x =>
                              match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                              | of x => WithInitial.of (op x)
                              | star => WithInitial.star :
                            WithInitial Cᵒᵖ))
                  x, y, f, x_3 with
                | op (of x), op (of y), f, x_4 => (down f).op
                | op star, op (of a), x, x_4 =>
                  Limits.IsInitial.to WithInitial.starInitial
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op (of a) with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star)
                | op star, op star, x, x_4 =>
                  𝟙
                    (match (motive := (WithTerminal C)ᵒᵖ → WithInitial Cᵒᵖ) op star with
                    | op x =>
                      match (motive := WithTerminal C → WithInitial Cᵒᵖ) x with
                      | of x => WithInitial.of (op x)
                      | star => WithInitial.star),
            map_id := ⋯, map_comp := ⋯ }
          (op (of x))))

Tactic:
simp only [op_unop, Functor.id_obj, Functor.comp_obj, NatIso.ofComponents_hom_app,
          Iso.refl_hom, Category.comp_id]

Premises used: CategoryTheory.Functor.comp_obj, Opposite.op_unop, CategoryTheory.Category.comp_id, CategoryTheory.Functor.id_obj, CategoryTheory.Iso.refl_hom, CategoryTheory.NatIso.ofComponents_hom_app