-- 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_1 y₁ y₂ : D
f : y₁ ⟶ y₂
x y : C
a :
  Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_5)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x_1) x)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y₁) y) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (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 _) y₂) _)
        (tensorObj.{v₁, u₁} (𝒞 := inst_4) x y))
    (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x_1) x ◁
        NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f) y ≫
      NatTrans.app
        (unit (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 _) y₂))
        (x, y))
    (NatTrans.app
        (unit (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 _) y₁))
        (x, y) ≫
      NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (x_1 ◁ f))
        (tensorObj.{v₁, u₁} (𝒞 := inst_4) x y))
⊢ Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_5)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x_1) x)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y₁) y) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (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 _) y₂) _)
        (tensorObj.{v₁, u₁} (𝒞 := inst_4) x y))
    (CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (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 _) y₁) _)
          (tensorObj.{v₁, u₁} (𝒞 := inst_4) x y))
        (NatTrans.app
          (unit (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 _) y₁))
          (x, y))
        (NatTrans.app
          (map (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 x_1))
            (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f))
          (tensorObj.{v₁, u₁} (𝒞 := inst_4) x y)) :
      tensorObj.{v₂, u₂} (𝒞 := inst_5)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x_1) x)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y₁) y) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (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 _) y₂) _)
          (tensorObj.{v₁, u₁} (𝒞 := inst_4) x y))
    (NatTrans.app
          (unit (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 _) y₁))
          (x, y) ≫
        NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (x_1 ◁ f))
          (tensorObj.{v₁, u₁} (𝒞 := inst_4) x y) :
      tensorObj.{v₂, u₂} (𝒞 := inst_5)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x_1) x)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y₁) y) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (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 _) y₂) _)
          (tensorObj.{v₁, u₁} (𝒞 := inst_4) x y))

Tactic:
simp only [Functor.map_id, unit_app_map_app, Functor.comp_obj, tensor_obj,
        NatTrans.id_app, id_tensorHom]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.MonoidalCategory.id_tensorHom, CategoryTheory.Functor.map_id, CategoryTheory.NatTrans.id_app, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_map_app