-- 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 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_10 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_12 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d)
      (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d)
      (tensorLeft v)
this : MonoidalCategoryStruct.{v₃, u₃} D := mkMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d₁ d₂ d₃ : D
x₁ x₂ x₃ : C
simp :
  Eq.{v₂ + 1} (α :=
    MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
        (MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) x₂))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
          (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁
            (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃)))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂) x₃))
    (CategoryStruct.comp (Y :=
        MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
          (Functor.obj.{v₁, v₂, u₁, u₂}
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
              (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁ d₂))
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂))
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃))
        (NatTrans.app
            (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂))
            (x₁, x₂) ▷
          Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃)
        (CategoryStruct.comp (Y :=
          Functor.obj.{v₁, v₂, u₁, u₂}
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
              (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8)
                (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁ d₂) d₃))
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4)
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂) x₃))
          (NatTrans.app
            (DayConvolution.unit
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁ d₂))
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃))
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂, x₃))
          (NatTrans.app
            (Iso.hom
              (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁)
                (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃)))
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4)
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂) x₃))) :
      MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
          (MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) x₂))
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
            (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁
              (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃)))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4)
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂) x₃))
    (Iso.hom
          (α_ (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) x₂)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃)) ≫
        CategoryStruct.comp (Y :=
          MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
            (Functor.obj.{v₁, v₂, u₁, u₂}
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃))
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₂ x₃)))
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁ ◁
              NatTrans.app
                (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂)
                  (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃))
                (x₂, x₃) :
            MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
                (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
                (MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
                  (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) x₂)
                  (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃)) ⟶
              MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
                (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
                (Functor.obj.{v₁, v₂, u₁, u₂}
                  (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                    (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃))
                  (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₂ x₃)))
          (NatTrans.app
                (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁)
                  (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                    (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃)))
                (x₁, MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₂ x₃) ≫
              Functor.map
                (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                  (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁
                    (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃)))
                (Iso.inv (α_ x₁ x₂ x₃)) :
            MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
                (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
                (Functor.obj.{v₁, v₂, u₁, u₂}
                  (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                    (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃))
                  (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₂ x₃)) ⟶
              Functor.obj.{v₁, v₂, u₁, u₂}
                (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                  (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁
                    (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃)))
                (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4)
                  (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂) x₃)) :
      MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
          (MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) x₂))
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
            (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁
              (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃)))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4)
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂) x₃))
⊢ Eq.{v₂ + 1} (α :=
    MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
        (Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁ ⊠ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂) (x₁, x₂))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
          (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁
            (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃)))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂) x₃))
    (CategoryStruct.comp (Y :=
      MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
        (Functor.obj.{v₁, v₂, u₁, u₂}
          (tensor C ⋙
            Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
              (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁ d₂))
          (x₁, x₂))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃))
      (NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂))
          (x₁, x₂) ▷
        Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃) x₃)
      (NatTrans.app
          (DayConvolution.unit
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
              (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁ d₂))
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂, x₃) ≫
        NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ _) (Iso.hom (α_ d₁ d₂ d₃)))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4)
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₁ x₂) x₃)))
    (Iso.hom
        (α_ (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
          (Functor.obj.{v₁, v₂, u₁, u₂}
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂, Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃).1
            (x₂, x₃).1)
          (Functor.obj.{v₁, v₂, u₁, u₂}
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂, Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃).2
            (x₂, x₃).2)) ≫
      CategoryStruct.comp (Y :=
        MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁)
          (Functor.obj.{v₁, v₂, u₁, u₂}
            (tensor C ⋙
              Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃))
            (x₂, x₃)))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁) x₁ ◁
          NatTrans.app
            (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₂)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₃))
            (x₂, x₃))
        (NatTrans.app
            (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) d₁)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃)))
            (x₁, MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₂ x₃) ≫
          Functor.map
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
              (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₁
                (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) d₂ d₃)))
            (Iso.inv (α_ (x₁, MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x₂ x₃).1 x₂ x₃))))

Tactic:
simp only [externalProductBifunctor_obj_obj, Functor.comp_obj, tensor_obj, associator,
        Functor.FullyFaithful.preimageIso_hom, Functor.FullyFaithful.map_preimage]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.MonoidalCategory.externalProductBifunctor_obj_obj, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.Functor.FullyFaithful.preimageIso_hom, CategoryTheory.Functor.FullyFaithful.map_preimage, CategoryTheory.MonoidalCategoryStruct.associator