-- 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
inst_4 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_5 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_6 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        d)
      (tensorRight v)
F G U : Functor.{v₁, v₂, u₁, u₂} C V
inst_9 : DayConvolutionUnit U
inst_10 : DayConvolution F U
inst_11 : DayConvolution U G
inst_12 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 U G _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F U _) G
inst_14 : DayConvolution F G
this :
  ((F ⊠ U) ⊠ G).IsLeftKanExtension (L :=
    Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C)) (F :=
    (F ⊠ Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) ⊠ G)
    (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G)
x : (C × Discrete PUnit.{1}) × C
hγ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
        (tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) (G.obj x.2)) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 F G _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))) x.2))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
        (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2)))
      (F.obj x.1.1 ◁ (Iso.hom (λ_ (G.obj x.2)) ≫ G.map (Iso.inv (λ_ x.2))))
      (NatTrans.app (unit F G) (x.1.1, tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2) ≫
        (@convolution _ _ _ _ inst_2 inst_3 F G _).map (Iso.inv (α_ x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2))))
    (Iso.inv (α_ (F.obj x.1.1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) (G.obj x.2)) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (G.obj x.2))
        ((Iso.hom (ρ_ (F.obj x.1.1)) ≫ F.map (Iso.inv (ρ_ x.1.1))) ▷ G.obj x.2)
        (NatTrans.app (unit F G) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)), x.2)))
⊢ Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
        (tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) (G.obj x.2)) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 F G _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))) x.2))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
        (tensorObj.{v₂, u₂} (𝒞 := inst_1) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (G.obj x.2)))
      (F.obj x.1.1 ◁ can ▷ G.obj x.2)
      (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
          ((@convolution _ _ _ _ inst_2 inst_3 U G _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2)))
        (F.obj x.1.1 ◁ NatTrans.app (unit U G) (tensorUnit.{v₁, u₁} C (𝒞 := inst), x.2))
        (F.obj x.1.1 ◁
            NatTrans.app (Iso.hom (DayConvolutionUnit.leftUnitor U G))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2) ≫
          NatTrans.app (unit F G) (x.1.1, tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2) ≫
            (@convolution _ _ _ _ inst_2 inst_3 F G _).map
              (Iso.inv (α_ x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2)))))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
        (tensorObj.{v₂, u₂} (𝒞 := inst_1) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (G.obj x.2)))
      (F.obj x.1.1 ◁ can ▷ G.obj x.2)
      (Iso.inv (α_ (F.obj x.1.1) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (G.obj x.2)) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v₂, u₂} (𝒞 := inst_1)
            ((@convolution _ _ _ _ inst_2 inst_3 F U _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
            (G.obj x.2))
          (NatTrans.app (unit F U) (x.1.1, tensorUnit.{v₁, u₁} C (𝒞 := inst)) ▷ G.obj x.2)
          (NatTrans.app (Iso.hom (DayConvolutionUnit.rightUnitor U F))
                (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ▷
              G.obj x.2 ≫
            NatTrans.app (unit F G) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)), x.2))))

Tactic:
simp only [← MonoidalCategory.whiskerLeft_comp_assoc, leftUnitor_hom_unit_app,
    associator_inv_naturality_middle_assoc, ← comp_whiskerRight_assoc, rightUnitor_hom_unit_app]

Premises used: CategoryTheory.MonoidalCategory.DayConvolutionUnit.rightUnitor_hom_unit_app, CategoryTheory.MonoidalCategory.associator_inv_naturality_middle_assoc, CategoryTheory.MonoidalCategory.DayConvolutionUnit.leftUnitor_hom_unit_app, CategoryTheory.MonoidalCategory.comp_whiskerRight_assoc, CategoryTheory.MonoidalCategory.whiskerLeft_comp_assoc