-- 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
x : D
x_1 : C
simp :
  Eq (α :=
    MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x) x_1)
        (MonoidalCategoryStruct.tensorUnit.{v₂, u₂} V (𝒞 := inst_5)) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x)
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x_1
          (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
    (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 _ _) x) x_1)
        (Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) (MonoidalCategoryStruct.tensorUnit.{v₃, u₃} D (𝒞 := inst_8)))
          (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
      (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x) x_1 ◁ DayConvolutionUnit.can)
      (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) x
              (MonoidalCategoryStruct.tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x_1
            (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
        (NatTrans.app
            (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                (MonoidalCategoryStruct.tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
            (x_1, MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4)) :
          Functor.obj.{v₁, v₂, u₁, u₂}
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x ⊠
                Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                  (MonoidalCategoryStruct.tensorUnit.{v₃, u₃} D (𝒞 := inst_8)))
              (x_1, MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4)) ⟶
            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) x
                    (MonoidalCategoryStruct.tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
              (x_1, MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4)))
        (NatTrans.app
            (Iso.hom
              (DayConvolutionUnit.rightUnitor
                (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                  (@tensorUnit.{v₁, v₂, v₃, u₁, u₂, u₃} C inst_4 V inst_5 inst_6 inst_7 D inst_8 _))
                (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x)))
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x_1
              (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) :
          Functor.obj.{v₁, v₂, u₁, u₂}
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _)
                (MonoidalCategoryStruct.tensorObj.{v₃, u₃} (𝒞 := inst_8) x
                  (MonoidalCategoryStruct.tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x_1
                (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) ⟶
            Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x)
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x_1
                (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))))
    (Iso.hom (ρ_ (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x) x_1)) ≫
      Functor.map (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x) (Iso.inv (ρ_ x_1)))
⊢ Eq (α :=
    MonoidalCategoryStruct.tensorObj.{v₂, u₂} (𝒞 := inst_5)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x) x_1)
        (MonoidalCategoryStruct.tensorUnit.{v₂, u₂} V (𝒞 := inst_5)) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x)
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x_1
          (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
    (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 _ _) x) x_1)
        (Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) (MonoidalCategoryStruct.tensorUnit.{v₃, u₃} D (𝒞 := inst_8)))
          (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
      (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x) x_1 ◁ DayConvolutionUnit.can)
      (CategoryStruct.comp (Y :=
        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) x
                (MonoidalCategoryStruct.tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
          (x_1, MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4)))
        (NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) (MonoidalCategoryStruct.tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
          (x_1, MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4)))
        (NatTrans.app (Functor.map (@ι C _ V _ inst_6 inst_7 D _ _) (Iso.hom (ρ_ x)))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst_4) x_1
            (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))))
    (Iso.hom (ρ_ (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x) x_1)) ≫
      Functor.map (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ _) x) (Iso.inv (ρ_ x_1)))

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

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