-- Module: Mathlib.CategoryTheory.WithTerminal.Basic
Goal:
C : Type u
inst : Category.{v, u} C
x y z : WithTerminal Cᵒᵖ
f_1 : x ⟶ y
g_1 : y ⟶ z
x_1 x_2 : WithTerminal Cᵒᵖ
f : x_1 ⟶ x_2
g : x_2 ⟶ WithTerminal.star
⊢ Eq (α :=
    Quiver.Hom (V := (WithInitial C)ᵒᵖ)
      (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) x_1 with
        | WithTerminal.of x => op (of (unop x))
        | WithTerminal.star => op star :
        (WithInitial C)ᵒᵖ)
      (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) WithTerminal.star with
        | WithTerminal.of x => op (of (unop x))
        | WithTerminal.star => op star :
        (WithInitial C)ᵒᵖ))
    (match (motive :=
        (x y : WithTerminal Cᵒᵖ) →
          (x ⟶ y) →
            Quiver.Hom (V := (WithInitial C)ᵒᵖ)
              (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) x with
                | WithTerminal.of x => op (of (unop x))
                | WithTerminal.star => op star :
                (WithInitial C)ᵒᵖ)
              (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) y with
                | WithTerminal.of x => op (of (unop x))
                | WithTerminal.star => op star :
                (WithInitial C)ᵒᵖ))
        x_1, WithTerminal.star, f ≫ g with
      | WithTerminal.of (op x), WithTerminal.of (op y), f => down f
      | WithTerminal.of (op unop), WithTerminal.star, x =>
        op
          (Limits.IsInitial.to starInitial
            (Opposite.unop
              (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) WithTerminal.of (op unop) with
              | WithTerminal.of x => op (of (Opposite.unop x))
              | WithTerminal.star => op star)))
      | WithTerminal.star, WithTerminal.star, x =>
        𝟙
          (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) WithTerminal.star with
          | WithTerminal.of x => op (of (unop x))
          | WithTerminal.star => op star) :
      Quiver.Hom (V := (WithInitial C)ᵒᵖ)
        (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) x_1 with
          | WithTerminal.of x => op (of (unop x))
          | WithTerminal.star => op star :
          (WithInitial C)ᵒᵖ)
        (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) WithTerminal.star with
          | WithTerminal.of x => op (of (unop x))
          | WithTerminal.star => op star :
          (WithInitial C)ᵒᵖ))
    (CategoryStruct.comp (Y :=
        match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) x_2 with
        | WithTerminal.of x => op (of (unop x))
        | WithTerminal.star => op star)
        (match (motive :=
          (x y : WithTerminal Cᵒᵖ) →
            (x ⟶ y) →
              Quiver.Hom (V := (WithInitial C)ᵒᵖ)
                (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) x with
                  | WithTerminal.of x => op (of (unop x))
                  | WithTerminal.star => op star :
                  (WithInitial C)ᵒᵖ)
                (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) y with
                  | WithTerminal.of x => op (of (unop x))
                  | WithTerminal.star => op star :
                  (WithInitial C)ᵒᵖ))
          x_1, x_2, f with
        | WithTerminal.of (op x), WithTerminal.of (op y), f => down f
        | WithTerminal.of (op unop), WithTerminal.star, x =>
          op
            (Limits.IsInitial.to starInitial
              (Opposite.unop
                (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) WithTerminal.of (op unop) with
                | WithTerminal.of x => op (of (Opposite.unop x))
                | WithTerminal.star => op star)))
        | WithTerminal.star, WithTerminal.star, x =>
          𝟙
            (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) WithTerminal.star with
            | WithTerminal.of x => op (of (unop x))
            | WithTerminal.star => op star))
        (match (motive :=
          (x y : WithTerminal Cᵒᵖ) →
            (x ⟶ y) →
              Quiver.Hom (V := (WithInitial C)ᵒᵖ)
                (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) x with
                  | WithTerminal.of x => op (of (unop x))
                  | WithTerminal.star => op star :
                  (WithInitial C)ᵒᵖ)
                (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) y with
                  | WithTerminal.of x => op (of (unop x))
                  | WithTerminal.star => op star :
                  (WithInitial C)ᵒᵖ))
          x_2, WithTerminal.star, g with
        | WithTerminal.of (op x), WithTerminal.of (op y), f => down f
        | WithTerminal.of (op unop), WithTerminal.star, x =>
          op
            (Limits.IsInitial.to starInitial
              (Opposite.unop
                (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) WithTerminal.of (op unop) with
                | WithTerminal.of x => op (of (Opposite.unop x))
                | WithTerminal.star => op star)))
        | WithTerminal.star, WithTerminal.star, x =>
          𝟙
            (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) WithTerminal.star with
            | WithTerminal.of x => op (of (unop x))
            | WithTerminal.star => op star)) :
      Quiver.Hom (V := (WithInitial C)ᵒᵖ)
        (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) x_1 with
          | WithTerminal.of x => op (of (unop x))
          | WithTerminal.star => op star :
          (WithInitial C)ᵒᵖ)
        (match (motive := WithTerminal Cᵒᵖ → (WithInitial C)ᵒᵖ) WithTerminal.star with
          | WithTerminal.of x => op (of (unop x))
          | WithTerminal.star => op star :
          (WithInitial C)ᵒᵖ))

Tactic:
rfl