-- Module: Mathlib.CategoryTheory.WithTerminal.Basic
Goal:
C : Type u
inst : Category.{v, u} C
x : (WithTerminal C)ᵒᵖ
⊢ Eq (α :=
    Quiver.Hom (V := WithInitial Cᵒᵖ)
      (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 :
        WithInitial Cᵒᵖ)
      (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 :
        WithInitial Cᵒᵖ))
    (match (motive :=
        Quiver.Hom.{v, u} (V := (WithTerminal C)ᵒᵖ) (op star) (op star) →
          Quiver.Hom (V := WithInitial Cᵒᵖ)
            (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 :
              WithInitial Cᵒᵖ)
            (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 :
              WithInitial Cᵒᵖ))
        𝟙 (op star) 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))
          op star, op star, f, 𝟙 (op star) with
        | op (of x), op (of y), f, x_1 => (down f).op
        | op star, op (of a), x, x_1 =>
          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_1 =>
          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) :
      Quiver.Hom (V := WithInitial Cᵒᵖ)
        (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 :
          WithInitial Cᵒᵖ)
        (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 :
          WithInitial Cᵒᵖ))
    (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) :
      Quiver.Hom (V := WithInitial Cᵒᵖ)
        (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 :
          WithInitial Cᵒᵖ)
        (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 :
          WithInitial Cᵒᵖ))

Tactic:
rfl