-- 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 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorRight v)
inst_9 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorLeft v)
d₁ d₁' d₂ d₂' : D
f : d₁ ⟶ d₂
f' : d₁' ⟶ d₂'
u₁ u₂ : C
a :
  Eq.{v₂ + 1} (α :=
    MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁) u₁)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁') u₂) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
          (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂'))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂))
    (NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁'))
          (u₁, u₂) ≫
        NatTrans.app (Iso.hom (tensorObjIsoConvolution C V d₁ d₁'))
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂) ≫
          NatTrans.app
              (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
                (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f'))
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂) ≫
            NatTrans.app (Iso.inv (tensorObjIsoConvolution C V d₂ d₂'))
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂) :
      MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁) u₁)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁') u₂) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
            (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂'))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂))
    ((NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f) u₁ ⊗ₘ
          NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f') u₂) ≫
        NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₂)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₂'))
          (u₁, u₂) :
      MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁) u₁)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁') u₂) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₂)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₂') _)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂))
⊢ Eq.{v₂ + 1} (α :=
    MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁) u₁)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁') u₂) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
          (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂'))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂))
    (NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁'))
          (u₁, u₂) ≫
        NatTrans.app (Iso.hom (tensorObjIsoConvolution C V d₁ d₁'))
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂) ≫
          NatTrans.app
              (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
                (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f'))
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂) ≫
            NatTrans.app (Iso.inv (tensorObjIsoConvolution C V d₂ d₂'))
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂) :
      MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁) u₁)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁') u₂) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
            (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂'))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂))
    (CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁') _)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂))
        (NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁'))
          (u₁, u₂))
        (NatTrans.app
          (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
            (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f'))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂)) :
      MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁) u₁)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁') u₂) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
            (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂'))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) u₁ u₂))

Tactic:
simp only [DayConvolution.unit_app_map_app, Functor.comp_obj, tensor_obj]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_map_app