-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : MonoidalCategoryStruct.{v₃, u₃} D
inst_6 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d d' d'' : D
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_8 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
z x y : C
a :
  Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') y))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d'')))
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
    (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂}
            (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _)
            (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z))
        (NatTrans.app
              (unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'))
              (x, y) ▷
            Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z :
          tensorObj.{v₂, u₂} (𝒞 := inst_1)
              (tensorObj.{v₂, u₂} (𝒞 := inst_1)
                (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
                (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') y))
              (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z) ⟶
            tensorObj.{v₂, u₂} (𝒞 := inst_1)
              (Functor.obj.{v₁, v₂, u₁, u₂}
                (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _)
                (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
              (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z))
        (NatTrans.app
              (unit
                (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d''))
              (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z) ≫
            NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (α_ d d' d'')))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z) :
          tensorObj.{v₂, u₂} (𝒞 := inst_1)
              (Functor.obj.{v₁, v₂, u₁, u₂}
                (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _)
                (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
              (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z) ⟶
            Functor.obj.{v₁, v₂, u₁, u₂}
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
                (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d'')))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z)) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (tensorObj.{v₂, u₂} (𝒞 := inst_1)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') y))
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
            (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d'')))
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
    (Iso.hom
          (α_ (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') y)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z)) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v₂, u₂} (𝒞 := inst_1)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
            (Functor.obj.{v₁, v₂, u₁, u₂}
              (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') _)
              (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x ◁
              NatTrans.app
                (unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d''))
                (y, z) :
            tensorObj.{v₂, u₂} (𝒞 := inst_1)
                (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
                (tensorObj.{v₂, u₂} (𝒞 := inst_1)
                  (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') y)
                  (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z)) ⟶
              tensorObj.{v₂, u₂} (𝒞 := inst_1)
                (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
                (Functor.obj.{v₁, v₂, u₁, u₂}
                  (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') _)
                  (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
          (NatTrans.app
                (unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                  (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') _))
                (x, tensorObj.{v₁, u₁} (𝒞 := inst) y z) ≫
              Functor.map
                (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                  (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') _)
                  _)
                (Iso.inv (α_ x y z)) :
            tensorObj.{v₂, u₂} (𝒞 := inst_1)
                (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
                (Functor.obj.{v₁, v₂, u₁, u₂}
                  (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') _)
                  (tensorObj.{v₁, u₁} (𝒞 := inst) y z)) ⟶
              Functor.obj.{v₁, v₂, u₁, u₂}
                (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                  (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
                    (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') _)
                  _)
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z)) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (tensorObj.{v₂, u₂} (𝒞 := inst_1)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
            (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') y))
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') z) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
            (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'') _)
            _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
⊢ Eq.{v₂ + 1} (α :=
    Functor.obj.{v₁, v₂, u₁, u₂}
        ((Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d ⊠
            Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') ⊠
          Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
        ((x, y), z) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.prod (tensor C) (𝟭 C) ⋙
          tensor C ⋙
            Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
              (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d'')))
        ((x, y), z))
    (NatTrans.app
      ((Functor.homEquivOfIsLeftKanExtension
            (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _ ⊠
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
            (ExternalProduct.extensionUnitLeft
              (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _)
              (unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d''))
            (tensor C ⋙
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
                (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))) :
          (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
            ((ι C V D).obj d ⊠ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶
              (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
        ((Functor.CorepresentableBy.homEquiv
              (corepresentableBy
                (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')) :
            (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊛ (ι C V D).obj d'' ⟶ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
              ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
          (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (α_ d d' d'')))))
      ((x, y), z))
    (NatTrans.app
      ((Functor.homEquivOfIsLeftKanExtension
            (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _ ⊠
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
            (ExternalProduct.extensionUnitLeft
              (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _)
              (unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d''))
            (tensor C ⋙
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
                (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))) :
          (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
            ((ι C V D).obj d ⊠ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶
              (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
        ((Functor.CorepresentableBy.homEquiv
              (corepresentableBy
                (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') _)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')) :
            (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊛ (ι C V D).obj d'' ⟶ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
              ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
          (Iso.hom
            (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')))))
      ((x, y), z))

Tactic:
simp only [externalProductBifunctor_obj_obj, Functor.comp_obj, Functor.prod_obj,
    tensor_obj, Functor.id_obj, Functor.homEquivOfIsLeftKanExtension_apply_app,
    externalProductBifunctor_map_app, Functor.leftUnitor_inv_app, whiskerLeft_id,
    Category.comp_id, corepresentableBy_homEquiv_apply_app,
    DayConvolution.associator_hom_unit_unit]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.MonoidalCategory.externalProductBifunctor_obj_obj, CategoryTheory.MonoidalCategory.externalProductBifunctor_map_app, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.Functor.prod_obj, CategoryTheory.Category.comp_id, CategoryTheory.Functor.leftUnitor_inv_app, CategoryTheory.MonoidalCategory.whiskerLeft_id, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy_homEquiv_apply_app, CategoryTheory.Functor.id_obj, CategoryTheory.MonoidalCategory.DayConvolution.associator_hom_unit_unit, CategoryTheory.Functor.homEquivOfIsLeftKanExtension_apply_app