-- Mathlib/CategoryTheory/Limits/Presheaf.lean
@[reassoc (attr := simp)]
lemma coconeApp_naturality {P : Cᵒᵖ ⥤ Type v₁} {x y : P.Elements} (f : x ⟶ y) :
    yoneda.map f.1.unop ≫ coconeApp φ x = coconeApp φ y

Proof state:
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₁, u₂} D
F : C ⥤ D
G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁
φ : F ⋙ yoneda ⟶ yoneda ⋙ G
P : Cᵒᵖ ⥤ Type v₁
x y : P.Elements
f : x ⟶ y
eq₁ : yoneda.map (↑f).unop ≫ yonedaEquiv.symm x.snd = yonedaEquiv.symm y.snd
eq₂ :
  (G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop y.fst)))
      ((G.obj (yoneda.obj (Opposite.unop x.1))).map (F.map (↑f).unop).op
        ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst)))
          (𝟙 (F.obj (Opposite.unop x.fst))))) =
    (G.obj P).map (F.map (↑f).unop).op
      ((G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop x.fst)))
        ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst)))
          (𝟙 (F.obj (Opposite.unop x.fst)))))
eq₃ :
  (φ.app (Opposite.unop x.fst)).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst)))
      (𝟙 (F.toPrefunctor.1 (Opposite.unop y.fst)) ≫ F.map (↑f).unop) =
    (G.map (yoneda.map (↑f).unop)).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst)))
      ((φ.app (Opposite.unop y.fst)).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst)))
        (𝟙 (F.toPrefunctor.1 (Opposite.unop y.fst))))
eq₄ :
  ∀ (a : F.obj (Opposite.unop x.fst) ⟶ F.obj (Opposite.unop x.fst)),
    (φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop y.fst))) (F.map (↑f).unop ≫ a) =
      (G.obj (yoneda.obj (Opposite.unop x.fst))).map (F.map (↑f).unop).op
        ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst))) a)
⊢ yonedaEquiv (yoneda.map (↑f).unop ≫ coconeApp φ x) = yonedaEquiv (coconeApp φ y)

Tactic:
dsimp only [coconeApp]

Result:
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₁, u₂} D
F : C ⥤ D
G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁
φ : F ⋙ yoneda ⟶ yoneda ⋙ G
P : Cᵒᵖ ⥤ Type v₁
x y : P.Elements
f : x ⟶ y
eq₁ : yoneda.map (↑f).unop ≫ yonedaEquiv.symm x.snd = yonedaEquiv.symm y.snd
eq₂ :
  (G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop y.fst)))
      ((G.obj (yoneda.obj (Opposite.unop x.1))).map (F.map (↑f).unop).op
        ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst)))
          (𝟙 (F.obj (Opposite.unop x.fst))))) =
    (G.obj P).map (F.map (↑f).unop).op
      ((G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop x.fst)))
        ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst)))
          (𝟙 (F.obj (Opposite.unop x.fst)))))
eq₃ :
  (φ.app (Opposite.unop x.fst)).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst)))
      (𝟙 (F.toPrefunctor.1 (Opposite.unop y.fst)) ≫ F.map (↑f).unop) =
    (G.map (yoneda.map (↑f).unop)).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst)))
      ((φ.app (Opposite.unop y.fst)).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst)))
        (𝟙 (F.toPrefunctor.1 (Opposite.unop y.fst))))
eq₄ :
  ∀ (a : F.obj (Opposite.unop x.fst) ⟶ F.obj (Opposite.unop x.fst)),
    (φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop y.fst))) (F.map (↑f).unop ≫ a) =
      (G.obj (yoneda.obj (Opposite.unop x.fst))).map (F.map (↑f).unop).op
        ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst))) a)
⊢ yonedaEquiv
      (yoneda.map (↑f).unop ≫
        yonedaEquiv.symm
          ((G.map (yonedaEquiv.symm x.snd)).app (F.op.obj (Opposite.op (Opposite.unop x.fst)))
            ((φ.app (Opposite.unop x.fst)).app (F.op.obj (Opposite.op (Opposite.unop x.fst)))
              (𝟙 (Opposite.unop (F.op.obj (Opposite.op (Opposite.unop x.fst)))))))) =
    yonedaEquiv
      (yonedaEquiv.symm
        ((G.map (yonedaEquiv.symm y.snd)).app (F.op.obj (Opposite.op (Opposite.unop y.fst)))
          ((φ.app (Opposite.unop y.fst)).app (F.op.obj (Opposite.op (Opposite.unop y.fst)))
            (𝟙 (Opposite.unop (F.op.obj (Opposite.op (Opposite.unop y.fst))))))))

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₁, u₂} D
F : C ⥤ D
G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁
φ : F ⋙ yoneda ⟶ yoneda ⋙ G
P : Cᵒᵖ ⥤ Type v₁
x y : P.Elements
f : x ⟶ y
⊢ yoneda.map (↑f).unop ≫ coconeApp φ x = coconeApp φ y

Tactic:
have eq₁ : yoneda.map f.1.unop ≫ yonedaEquiv.symm x.2 = yonedaEquiv.symm y.2 :=
  yonedaEquiv.injective
    (by simpa only [Equiv.apply_symm_apply, ← yonedaEquiv_naturality] using f.2)

Result:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₁, u₂} D
F : C ⥤ D
G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁
φ : F ⋙ yoneda ⟶ yoneda ⋙ G
P : Cᵒᵖ ⥤ Type v₁
x y : P.Elements
f : x ⟶ y
eq₁ : yoneda.map (↑f).unop ≫ yonedaEquiv.symm x.snd = yonedaEquiv.symm y.snd
⊢ yoneda.map (↑f).unop ≫ coconeApp φ x = coconeApp φ y