-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (goal)
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : MonoidalCategoryStruct.{v₃, u₃} D
inst_10 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_12 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorLeft v)
inst_14 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorRight v)
inst_15 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
        d)
      (tensorRight v)
inst_16 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
x y : D
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} (C × C) V)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x ⊠ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)
      (tensor C ⋙
        @DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y) _))
    ((Functor.CorepresentableBy.homEquiv
            (corepresentableBy (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)) :
          (a : (ι C V D).obj x ⊛ (ι C V D).obj y ⟶ (ι C V D).obj x ⊛ (ι C V D).obj y) →
            (ι C V D).obj x ⊠ (ι C V D).obj y ⟶ tensor C ⋙ (ι C V D).obj x ⊛ (ι C V D).obj y)
        (map (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x))
          (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y))) :
      Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} (C × C) V)
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x ⊠ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)
        (tensor C ⋙
          @DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y) _))
    ((Functor.CorepresentableBy.homEquiv
            (corepresentableBy (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)) :
          (a : (ι C V D).obj x ⊛ (ι C V D).obj y ⟶ (ι C V D).obj x ⊛ (ι C V D).obj y) →
            (ι C V D).obj x ⊠ (ι C V D).obj y ⟶ tensor C ⋙ (ι C V D).obj x ⊛ (ι C V D).obj y)
        (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y))) :
      Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} (C × C) V)
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x ⊠ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)
        (tensor C ⋙
          @DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y) _))
have a : ∀ (fst snd : C),
  Eq.{v₂ + 1} (α :=
    Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x ⊠ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)
        (fst, snd) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (tensor C ⋙
          @DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y) _)
        (fst, snd))
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv
            (corepresentableBy (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)) :
          (a : (ι C V D).obj x ⊛ (ι C V D).obj y ⟶ (ι C V D).obj x ⊛ (ι C V D).obj y) →
            (ι C V D).obj x ⊠ (ι C V D).obj y ⟶ tensor C ⋙ (ι C V D).obj x ⊛ (ι C V D).obj y)
        (map (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x))
          (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y))))
      (fst, snd))
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv
            (corepresentableBy (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)) :
          (a : (ι C V D).obj x ⊛ (ι C V D).obj y ⟶ (ι C V D).obj x ⊛ (ι C V D).obj y) →
            (ι C V D).obj x ⊠ (ι C V D).obj y ⟶ tensor C ⋙ (ι C V D).obj x ⊛ (ι C V D).obj y)
        (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y))))
      (fst, snd))
Tactic: ext ⟨_, _⟩