-- 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
H : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution G H
inst_6 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 G H _)
inst_7 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F G _) H
inst_8 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_9 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
x y z : C
this :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
    (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          ((@convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y)) (H.obj z))
        (NatTrans.app (unit F G) (x, y) ▷ H.obj z ≫
          (@convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y) ◁ 𝟙 (H.obj z))
        (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H) (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z) ≫
          NatTrans.app
            ((@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H
                  _).descOfIsLeftKanExtension
              (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H)
              (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
              ((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).descOfIsLeftKanExtension
                (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
                (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
                (((Equiv.symm
                        (FullyFaithful.homEquiv
                          (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.associativity C C C)))) :
                      (prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                          prod.inverseAssociator C C C ⋙
                            Functor.prod (tensor C) (𝟭 C) ⋙
                              tensor C ⋙
                                @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) ≃
                        ((F ⊠ G) ⊠ H ⟶
                          Functor.prod (tensor C) (𝟭 C) ⋙
                            tensor C ⋙
                              @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) :
                    (a :
                        prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                          prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H) →
                      (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
                  ((ConcreteCategory.hom (FC := TypeCat.Fun)
                        (𝟙
                          (prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                            prod.inverseAssociator C C C ⋙
                              Functor.prod (tensor C) (𝟭 C) ⋙
                                tensor C ⋙
                                  @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) :
                      (a :
                          prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                            (prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C) ⋙ F ⊛ G ⊛ H) →
                        prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                          prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
                    (CategoryStruct.comp (Y :=
                      (Functor.prod (𝟭 C) (tensor C) ⋙ tensor C) ⋙
                        @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
                      (CategoryStruct.comp (Y := F ⊠ G ⊠ H)
                        (Iso.inv
                          (NatIso.ofComponents
                            (fun (x : C × C × C) ↦ Iso.symm (α_ (F.obj x.1) (G.obj x.2.1) (H.obj x.2.2))) ⋯))
                        (((F ⊠ @convolution _ _ _ _ inst_2 inst_3 G H _).homEquivOfIsLeftKanExtension
                              (extensionUnitRight (@convolution _ _ _ _ inst_2 inst_3 G H _) (unit G H) F)
                              (tensor C ⋙
                                @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) :
                            (a : F ⊠ G ⊛ H ⟶ tensor C ⋙ F ⊛ G ⊛ H) →
                              F ⊠ G ⊠ H ⟶ (𝟭 C).prod (tensor C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
                          ((CorepresentableBy.homEquiv
                                (corepresentableBy F (@convolution _ _ _ _ inst_2 inst_3 G H _)) :
                              (a : F ⊛ G ⊛ H ⟶ F ⊛ G ⊛ H) → F ⊠ G ⊛ H ⟶ tensor C ⋙ F ⊛ G ⊛ H)
                            (𝟙 (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)))))
                      (NatTrans.app
                        (Functor.map (whiskeringLeft (C × C × C) C V)
                          (Iso.inv (NatIso.ofComponents (fun (x : C × C × C) ↦ α_ x.1 x.2.1 x.2.2) ⋯)))
                        (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)))))))
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z)) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
        (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
    (NatTrans.app
        (((Equiv.symm
                (FullyFaithful.homEquiv
                  (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.associativity C C C)))) :
              (prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                  prod.inverseAssociator C C C ⋙
                    Functor.prod (tensor C) (𝟭 C) ⋙
                      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) ≃
                ((F ⊠ G) ⊠ H ⟶
                  Functor.prod (tensor C) (𝟭 C) ⋙
                    tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) :
            (a :
                prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                  prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H) →
              (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
          ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (𝟙
                  (prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                    prod.inverseAssociator C C C ⋙
                      Functor.prod (tensor C) (𝟭 C) ⋙
                        tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) :
              (a :
                  prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                    (prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C) ⋙ F ⊛ G ⊛ H) →
                prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                  prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
            (CategoryStruct.comp (Y :=
              (Functor.prod (𝟭 C) (tensor C) ⋙ tensor C) ⋙
                @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
              (CategoryStruct.comp (Y := F ⊠ G ⊠ H)
                (Iso.inv
                  (NatIso.ofComponents (fun (x : C × C × C) ↦ Iso.symm (α_ (F.obj x.1) (G.obj x.2.1) (H.obj x.2.2))) ⋯))
                (((F ⊠ @convolution _ _ _ _ inst_2 inst_3 G H _).homEquivOfIsLeftKanExtension
                      (extensionUnitRight (@convolution _ _ _ _ inst_2 inst_3 G H _) (unit G H) F)
                      (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) :
                    (a : F ⊠ G ⊛ H ⟶ tensor C ⋙ F ⊛ G ⊛ H) → F ⊠ G ⊠ H ⟶ (𝟭 C).prod (tensor C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
                  ((CorepresentableBy.homEquiv (corepresentableBy F (@convolution _ _ _ _ inst_2 inst_3 G H _)) :
                      (a : F ⊛ G ⊛ H ⟶ F ⊛ G ⊛ H) → F ⊠ G ⊛ H ⟶ tensor C ⋙ F ⊛ G ⊛ H)
                    (𝟙 (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)))))
              (NatTrans.app
                (Functor.map (whiskeringLeft (C × C × C) C V)
                  (Iso.inv (NatIso.ofComponents (fun (x : C × C × C) ↦ α_ x.1 x.2.1 x.2.2) ⋯)))
                (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)))))
        ((x, y), z) :
      ((F ⊠ G) ⊠ H).obj ((x, y), z) ⟶
        (Functor.prod (tensor C) (𝟭 C) ⋙
              tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).obj
          ((x, y), z))
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
    (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          ((@convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y)) (H.obj z))
        (NatTrans.app (unit F G) (x, y) ▷ H.obj z)
        (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H) (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z) ≫
          NatTrans.app
            ((@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H
                  _).descOfIsLeftKanExtension
              (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H)
              (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
              ((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).descOfIsLeftKanExtension
                (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
                (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
                (((Equiv.symm
                        (FullyFaithful.homEquiv
                          (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.associativity C C C)))) :
                      (prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                          prod.inverseAssociator C C C ⋙
                            Functor.prod (tensor C) (𝟭 C) ⋙
                              tensor C ⋙
                                @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) ≃
                        ((F ⊠ G) ⊠ H ⟶
                          Functor.prod (tensor C) (𝟭 C) ⋙
                            tensor C ⋙
                              @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) :
                    (a :
                        prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                          prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H) →
                      (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
                  ((ConcreteCategory.hom (FC := TypeCat.Fun)
                        (𝟙
                          (prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                            prod.inverseAssociator C C C ⋙
                              Functor.prod (tensor C) (𝟭 C) ⋙
                                tensor C ⋙
                                  @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) :
                      (a :
                          prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                            (prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C) ⋙ F ⊛ G ⊛ H) →
                        prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                          prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
                    (CategoryStruct.comp (Y := F ⊠ G ⊠ H)
                      (Iso.inv
                        (NatIso.ofComponents
                          (fun (x : C × C × C) ↦ Iso.symm (α_ (F.obj x.1) (G.obj x.2.1) (H.obj x.2.2))) ⋯))
                      (CategoryStruct.comp (Y :=
                        (Functor.prod (𝟭 C) (tensor C) ⋙ tensor C) ⋙
                          @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
                        (((F ⊠ @convolution _ _ _ _ inst_2 inst_3 G H _).homEquivOfIsLeftKanExtension
                              (extensionUnitRight (@convolution _ _ _ _ inst_2 inst_3 G H _) (unit G H) F)
                              (tensor C ⋙
                                @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) :
                            (a : F ⊠ G ⊛ H ⟶ tensor C ⋙ F ⊛ G ⊛ H) →
                              F ⊠ G ⊠ H ⟶ (𝟭 C).prod (tensor C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
                          ((CorepresentableBy.homEquiv
                                (corepresentableBy F (@convolution _ _ _ _ inst_2 inst_3 G H _)) :
                              (a : F ⊛ G ⊛ H ⟶ F ⊛ G ⊛ H) → F ⊠ G ⊛ H ⟶ tensor C ⋙ F ⊛ G ⊛ H)
                            (𝟙 (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _))))
                        (NatTrans.app
                          (Functor.map (whiskeringLeft (C × C × C) C V)
                            (Iso.inv (NatIso.ofComponents (fun (x : C × C × C) ↦ α_ x.1 x.2.1 x.2.2) ⋯)))
                          (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _))))))))
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z)) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
        (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
    (NatTrans.app
        (((Equiv.symm
                (FullyFaithful.homEquiv
                  (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.associativity C C C)))) :
              (prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                  prod.inverseAssociator C C C ⋙
                    Functor.prod (tensor C) (𝟭 C) ⋙
                      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) ≃
                ((F ⊠ G) ⊠ H ⟶
                  Functor.prod (tensor C) (𝟭 C) ⋙
                    tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) :
            (a :
                prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                  prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H) →
              (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
          ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (𝟙
                  (prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                    prod.inverseAssociator C C C ⋙
                      Functor.prod (tensor C) (𝟭 C) ⋙
                        tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) :
              (a :
                  prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                    (prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C) ⋙ F ⊛ G ⊛ H) →
                prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H ⟶
                  prod.inverseAssociator C C C ⋙ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
            (CategoryStruct.comp (Y := F ⊠ G ⊠ H)
              (Iso.inv
                (NatIso.ofComponents (fun (x : C × C × C) ↦ Iso.symm (α_ (F.obj x.1) (G.obj x.2.1) (H.obj x.2.2))) ⋯))
              (CategoryStruct.comp (Y :=
                (Functor.prod (𝟭 C) (tensor C) ⋙ tensor C) ⋙
                  @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
                (((F ⊠ @convolution _ _ _ _ inst_2 inst_3 G H _).homEquivOfIsLeftKanExtension
                      (extensionUnitRight (@convolution _ _ _ _ inst_2 inst_3 G H _) (unit G H) F)
                      (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) :
                    (a : F ⊠ G ⊛ H ⟶ tensor C ⋙ F ⊛ G ⊛ H) → F ⊠ G ⊠ H ⟶ (𝟭 C).prod (tensor C) ⋙ tensor C ⋙ F ⊛ G ⊛ H)
                  ((CorepresentableBy.homEquiv (corepresentableBy F (@convolution _ _ _ _ inst_2 inst_3 G H _)) :
                      (a : F ⊛ G ⊛ H ⟶ F ⊛ G ⊛ H) → F ⊠ G ⊛ H ⟶ tensor C ⋙ F ⊛ G ⊛ H)
                    (𝟙 (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _))))
                (NatTrans.app
                  (Functor.map (whiskeringLeft (C × C × C) C V)
                    (Iso.inv (NatIso.ofComponents (fun (x : C × C × C) ↦ α_ x.1 x.2.1 x.2.2) ⋯)))
                  (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _))))))
        ((x, y), z) :
      ((F ⊠ G) ⊠ H).obj ((x, y), z) ⟶
        (Functor.prod (tensor C) (𝟭 C) ⋙
              tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).obj
          ((x, y), z))

Tactic:
simp only [whiskerLeft_id, Category.comp_id, Category.assoc] at this

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Category.comp_id, CategoryTheory.MonoidalCategory.whiskerLeft_id