-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
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
a :
  Eq (α :=
    Functor.obj (C := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙
        Functor.obj coyoneda
          (Opposite.op
            (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)))
      (@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) →
            ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  coyoneda.obj (Opposite.op ((ι C V D).obj x ⊠ (ι C V D).obj y))).obj
              ((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))) :
      Functor.obj (C := Functor.{v₁, v₂, u₁, u₂} C V)
        (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙
          Functor.obj coyoneda
            (Opposite.op
              (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)))
        (@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) →
            ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  coyoneda.obj (Opposite.op ((ι C V D).obj x ⊠ (ι C V D).obj y))).obj
              ((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))) :
      Functor.obj (C := Functor.{v₁, v₂, u₁, u₂} C V)
        (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙
          Functor.obj coyoneda
            (Opposite.op
              (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)))
        (@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) _))
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x 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 V)
        (@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) _)
        (@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) _))
    (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))

Tactic:
apply (DayConvolution.corepresentableBy
        (ι C V D |>.obj _) (ι C V D |>.obj _)).homEquiv.injective

Premises used: CategoryTheory.Functor.CorepresentableBy.homEquiv, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy, Equiv.injective, CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.ι, CategoryTheory.Functor.obj