-- 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
F G : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolution F G
inst_5 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_7 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
H K : Functor.{v₁, v₂, u₁, u₂} C V
inst_8 : DayConvolution G H
inst_9 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F G _) H
inst_10 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 G H _)
inst_11 : DayConvolution H K
inst_12 : DayConvolution G (@convolution _ _ _ _ inst_2 inst_3 H K _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 G H _) K
inst_14 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K
inst_15 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F G _) (@convolution _ _ _ _ inst_2 inst_3 H K _)
inst_16 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) K
inst_17 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 G (@convolution _ _ _ _ inst_2 inst_3 H K _) _)
inst_18 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 G H _) K _)
this :
  ((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H) ⊠ K).IsLeftKanExtension
    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H)
      (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H) K)
l k i j : C
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) ((F ⊠ G).obj (i, j)) ((H ⊠ K).obj (k, l)) ⟶
      (tensor C ⋙
            @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _)
              (@convolution _ _ _ _ inst_2 inst_3 H K _) _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) i j, tensorObj.{v₁, u₁} (𝒞 := inst) k l))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1) ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F G _).obj (i, j))
        ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 H K _).obj (k, l)))
      (NatTrans.app (unit F G) (i, j) ⊗ₘ NatTrans.app (unit H K) (k, l))
      (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) (@convolution _ _ _ _ inst_2 inst_3 H K _))
        (tensorObj.{v₁, u₁} (𝒞 := inst) i j, tensorObj.{v₁, u₁} (𝒞 := inst) k l)))
    (Iso.hom (α_ (F.obj i) (G.obj j) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (H.obj k) (K.obj l))) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj i)
          (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj j)
            ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 H K _).obj (k, l))))
        (F.obj i ◁ G.obj j ◁ NatTrans.app (unit H K) (k, l) :
          tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj i)
              (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj j) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (H.obj k) (K.obj l))) ⟶
            tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj i)
              (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj j)
                ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 H K _).obj (k, l))))
        (CategoryStruct.comp (Y :=
            tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj i)
              ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 G (@convolution _ _ _ _ inst_2 inst_3 H K _) _).obj
                (j, tensorObj.{v₁, u₁} (𝒞 := inst) k l)))
            (F.obj i ◁
                NatTrans.app (unit G (@convolution _ _ _ _ inst_2 inst_3 H K _))
                  (j, tensorObj.{v₁, u₁} (𝒞 := inst) k l) :
              tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj i)
                  (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj j)
                    ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 H K _).obj (k, l))) ⟶
                tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj i)
                  ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 G (@convolution _ _ _ _ inst_2 inst_3 H K _) _).obj
                    (j, tensorObj.{v₁, u₁} (𝒞 := inst) k l)))
            (NatTrans.app (unit F (@convolution _ _ _ _ inst_2 inst_3 G (@convolution _ _ _ _ inst_2 inst_3 H K _) _))
                  (i, tensorObj.{v₁, u₁} (𝒞 := inst) j (tensorObj.{v₁, u₁} (𝒞 := inst) k l)) ≫
                (@convolution _ _ _ _ inst_2 inst_3 F
                        (@convolution _ _ _ _ inst_2 inst_3 G (@convolution _ _ _ _ inst_2 inst_3 H K _) _) _).map
                    (Iso.inv (α_ i j (tensorObj.{v₁, u₁} (𝒞 := inst) k l))) ≫
                  NatTrans.app (Iso.inv (associator F G (@convolution _ _ _ _ inst_2 inst_3 H K _)))
                    (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) i j)
                      (tensorObj.{v₁, u₁} (𝒞 := inst) k l)) :
              tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj i)
                  ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 G (@convolution _ _ _ _ inst_2 inst_3 H K _) _).obj
                    (j, tensorObj.{v₁, u₁} (𝒞 := inst) k l)) ⟶
                (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _)
                      (@convolution _ _ _ _ inst_2 inst_3 H K _) _).obj
                  (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) i j)
                    (tensorObj.{v₁, u₁} (𝒞 := inst) k l))) :
          tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj i)
              (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj j)
                ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 H K _).obj (k, l))) ⟶
            (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _)
                  (@convolution _ _ _ _ inst_2 inst_3 H K _) _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) i j)
                (tensorObj.{v₁, u₁} (𝒞 := inst) k l))))

Tactic:
have aux :
      ((unit F G).app (i, j) ⊗ₘ (unit H K).app (k, l)) ≫
        (unit (F ⊛ G) (H ⊛ K)).app ((i ⊗ j), (k ⊗ l)) =
      (α_ (F.obj i) (G.obj j) (H.obj k ⊗ K.obj l)).hom ≫
        F.obj i ◁ G.obj j ◁ (unit H K).app (k, l) ≫ F.obj i ◁ (unit G (H ⊛ K)).app (j, (k ⊗ l)) ≫
        (unit F (G ⊛ H ⊛ K)).app (i, (j ⊗ k ⊗ l)) ≫ (F ⊛ G ⊛ H ⊛ K).map (α_ i j (k ⊗ l)).inv ≫
        (associator F G (H ⊛ K)).inv.app ((i ⊗ j) ⊗ k ⊗ l) := by
    conv_rhs => simp only [Functor.comp_obj, tensor_obj, NatTrans.naturality,
      associator_inv_unit_unit_assoc, externalProductBifunctor_obj_obj, Iso.map_hom_inv_id,
      Category.comp_id]
    simp only [tensor_whiskerLeft_symm, Category.assoc, Iso.hom_inv_id_assoc,
    ← tensorHom_def'_assoc]