-- Module: Mathlib.CategoryTheory.Monoidal.Types.Coyoneda
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : MonoidalCategory.{v, u} C
X : C
h :
  ∀ (f : Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X),
    TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (Iso.hom (ρ_ (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X))))
        (f, PUnit.unit) =
      TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryStruct.comp (Y :=
            tensorObj (𝒞 := types) (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X)
              (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst))))
                (tensorUnit.{v, u} C (𝒞 := inst))))
            (𝟙 (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X) ⊗ₘ
              ↾fun (x : (𝟙_ (Type v)) (𝒞 := types)) ↦ 𝟙 (unop (op (tensorUnit.{v, u} C (𝒞 := inst)))))
            (CategoryStruct.comp (Y :=
              Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst))))
                (tensorObj.{v, u} (𝒞 := inst) X (tensorUnit.{v, u} C (𝒞 := inst))))
              ((fun (X Y : C) ↦
                  (↾fun
                        (p :
                          tensorObj (𝒞 := types)
                            (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X)
                            (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) Y)) ↦
                      Iso.inv (λ_ (tensorUnit.{v, u} C (𝒞 := inst))) ≫ (p.1 ⊗ₘ p.2) :
                    tensorObj (𝒞 := types) (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X)
                        (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) Y) ⟶
                      Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst))))
                        (tensorObj.{v, u} (𝒞 := inst) X Y)))
                X (tensorUnit.{v, u} C (𝒞 := inst)))
              (Functor.map (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) (Iso.hom (ρ_ X))))))
        (f, PUnit.unit)
⊢ Eq (α :=
    tensorObj (𝒞 := types) (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X)
        ((𝟙_ (Type v)) (𝒞 := types)) ⟶
      Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X)
    (Iso.hom (ρ_ (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X)))
    (CategoryStruct.comp (Y :=
      tensorObj (𝒞 := types) (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X)
        (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) (tensorUnit.{v, u} C (𝒞 := inst))))
      (𝟙 (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X) ⊗ₘ
        ↾fun (x : (𝟙_ (Type v)) (𝒞 := types)) ↦ 𝟙 (unop (op (tensorUnit.{v, u} C (𝒞 := inst)))))
      (CategoryStruct.comp (Y :=
        Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst))))
          (tensorObj.{v, u} (𝒞 := inst) X (tensorUnit.{v, u} C (𝒞 := inst))))
        ((fun (X Y : C) ↦
            (↾fun
                  (p :
                    tensorObj (𝒞 := types) (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X)
                      (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) Y)) ↦
                Iso.inv (λ_ (tensorUnit.{v, u} C (𝒞 := inst))) ≫ (p.1 ⊗ₘ p.2) :
              tensorObj (𝒞 := types) (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) X)
                  (Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) Y) ⟶
                Functor.obj (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst))))
                  (tensorObj.{v, u} (𝒞 := inst) X Y)))
          X (tensorUnit.{v, u} C (𝒞 := inst)))
        (Functor.map (Functor.obj coyoneda (op (tensorUnit.{v, u} C (𝒞 := inst)))) (Iso.hom (ρ_ X)))))

Tactic:
ext ⟨f, ⟨⟩⟩