-- 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)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : D
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
a :
  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) (tensorObj.{v₃, u₃} (𝒞 := inst_8) X₁ X₂) X₃))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y₁ (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y₂ Y₃))))
    (CategoryStruct.comp (Y :=
        Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y₁ Y₂) Y₃))
        (map
          (map (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f₁)
            (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f₂))
          (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f₃))
        (Iso.hom
          (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) Y₁)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) Y₂)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) Y₃))) :
      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) (tensorObj.{v₃, u₃} (𝒞 := inst_8) X₁ X₂) X₃))
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y₁ (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y₂ Y₃))))
    (CategoryStruct.comp (Y :=
        Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_8) X₁ (tensorObj.{v₃, u₃} (𝒞 := inst_8) X₂ X₃)))
        (Iso.hom
          (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) X₁)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) X₂)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) X₃)))
        (map (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f₁)
          (map (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f₂)
            (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f₃))) :
      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) (tensorObj.{v₃, u₃} (𝒞 := inst_8) X₁ X₂) X₃))
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y₁ (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y₂ Y₃))))
⊢ 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) (tensorObj.{v₃, u₃} (𝒞 := inst_8) X₁ X₂) X₃))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y₁ (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y₂ Y₃))))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (((f₁ ⊗ₘ f₂) ⊗ₘ f₃) ≫ Iso.hom (α_ Y₁ Y₂ Y₃)))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (Iso.hom (α_ X₁ X₂ X₃) ≫ (f₁ ⊗ₘ f₂ ⊗ₘ f₃)))

Tactic:
simp only [Functor.map_comp, ι_map_associator_hom_eq_associator_hom,
        ι_map_tensorHom_hom_eq_tensorHom]

Premises used: CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.ι_map_associator_hom_eq_associator_hom, CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.ι_map_tensorHom_hom_eq_tensorHom, CategoryTheory.Functor.map_comp