-- 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.{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))))
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1) (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))
    (((𝟙 (F.obj x.1.1) ▷ tensorUnit.{v₂, u₂} V (𝒞 := inst_1) ≫ F.obj x.1.1 ◁ can) ▷ G.obj x.2 ≫
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ◁ 𝟙 (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 ≫
          (@convolution _ _ _ _ inst_2 inst_3 F U _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ◁
            𝟙 (G.obj x.2))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
              x.2))
          (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
            (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)), x.2))
          (CategoryStruct.comp (Y :=
            (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 U G _) _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
                x.2))
            (NatTrans.app (Iso.hom (associator F U G))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
                x.2))
            (NatTrans.app (map (𝟙 F) (Iso.hom (DayConvolutionUnit.leftUnitor U G)))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
                x.2)))))
    (((𝟙 (F.obj x.1.1) ▷ tensorUnit.{v₂, u₂} V (𝒞 := inst_1) ≫ F.obj x.1.1 ◁ can) ▷ G.obj x.2 ≫
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ◁ 𝟙 (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 ≫
          (@convolution _ _ _ _ inst_2 inst_3 F U _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ◁
            𝟙 (G.obj x.2))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
              x.2))
          (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
            (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)), x.2))
          (NatTrans.app (map (Iso.hom (DayConvolutionUnit.rightUnitor U F)) (𝟙 G))
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
              x.2))))

Tactic:
simp only [MonoidalCategory.whiskerRight_id, Category.id_comp, Iso.hom_inv_id, whisker_assoc,
    MonoidalCategory.whiskerLeft_id, Category.comp_id,
    DayConvolution.associator_hom_unit_unit_assoc, externalProductBifunctor_obj_obj, tensor_obj,
    NatTrans.naturality, unit_app_map_app_assoc, NatTrans.id_app, id_tensorHom,
    Category.assoc, Iso.inv_hom_id_assoc, unit_app_map_app, Functor.comp_obj,
    tensorHom_id, Iso.cancel_iso_hom_left]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.externalProductBifunctor_obj_obj, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.MonoidalCategory.id_tensorHom, CategoryTheory.Iso.hom_inv_id, CategoryTheory.Category.comp_id, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.Iso.cancel_iso_hom_left, CategoryTheory.MonoidalCategory.whiskerLeft_id, CategoryTheory.NatTrans.naturality, CategoryTheory.MonoidalCategory.whiskerRight_id, CategoryTheory.MonoidalCategory.DayConvolution.associator_hom_unit_unit_assoc, CategoryTheory.NatTrans.id_app, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_map_app_assoc, CategoryTheory.MonoidalCategory.whisker_assoc, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_map_app, CategoryTheory.MonoidalCategory.tensorHom_id, CategoryTheory.Category.id_comp