-- 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) (F.obj x) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z)) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
    (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x)
          ((@convolution _ _ _ _ inst_2 inst_3 G H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
        (F.obj x ◁ NatTrans.app (unit G H) (y, z))
        (NatTrans.app (unit F (@convolution _ _ _ _ inst_2 inst_3 G H _)) (x, tensorObj.{v₁, u₁} (𝒞 := inst) y z) ≫
          NatTrans.app
            ((@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _)
                  _).descOfIsLeftKanExtension
              (unit F (@convolution _ _ _ _ inst_2 inst_3 G H _))
              (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _)
              ((F ⊠ @convolution _ _ _ _ inst_2 inst_3 G H _).descOfIsLeftKanExtension
                (extensionUnitRight (@convolution _ _ _ _ inst_2 inst_3 G H _) (unit G H) F)
                (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _)
                (CategoryStruct.comp (Y := prod.inverseAssociator C C C ⋙ (F ⊠ G) ⊠ H)
                  (Iso.hom
                    (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 :=
                    (prod.inverseAssociator C C C ⋙ Functor.prod (tensor C) (𝟭 C) ⋙ tensor C) ⋙
                      @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 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 (@convolution _ _ _ _ inst_2 inst_3 F 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)
                      (((FullyFaithful.homEquiv
                              (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.associativity C C C))) :
                            ((F ⊠ G) ⊠ H ⟶
                                Functor.prod (tensor C) (𝟭 C) ⋙
                                  tensor C ⋙
                                    @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) ≃
                              (obj (obj (whiskeringLeft (C × C × C) ((C × C) × C) V) (prod.inverseAssociator C C C))
                                  ((F ⊠ G) ⊠ H) ⟶
                                obj (obj (whiskeringLeft (C × C × C) ((C × C) × C) V) (prod.inverseAssociator C C C))
                                  (Functor.prod (tensor C) (𝟭 C) ⋙
                                    tensor C ⋙
                                      @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H
                                        _))) :
                          (a : (F ⊠ G) ⊠ H ⟶ (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)
                        (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
                              (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
                              (tensor C ⋙
                                @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) :
                            (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ (F ⊛ G) ⊛ H) →
                              (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (F ⊛ G) ⊛ H)
                          ((CorepresentableBy.homEquiv
                                (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
                              (a : (F ⊛ G) ⊛ H ⟶ (F ⊛ G) ⊛ H) → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ (F ⊛ G) ⊛ H)
                            (𝟙 (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _))))))
                    (NatTrans.app
                      (Functor.map (whiskeringLeft (C × C × C) C V)
                        (Iso.hom (NatIso.ofComponents (fun (x : C × C × C) ↦ α_ x.1 x.2.1 x.2.2) ⋯)))
                      (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _))))))
            (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z))) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z)) ⟶
        (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
    (Iso.inv (α_ (F.obj x) (G.obj y) (H.obj z)) ≫
        NatTrans.app
            ((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 (@convolution _ _ _ _ inst_2 inst_3 F 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)
              (((FullyFaithful.homEquiv
                      (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.associativity C C C))) :
                    ((F ⊠ G) ⊠ H ⟶
                        Functor.prod (tensor C) (𝟭 C) ⋙
                          tensor C ⋙
                            @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) ≃
                      (obj (obj (whiskeringLeft (C × C × C) ((C × C) × C) V) (prod.inverseAssociator C C C))
                          ((F ⊠ G) ⊠ H) ⟶
                        obj (obj (whiskeringLeft (C × C × C) ((C × C) × C) V) (prod.inverseAssociator C C C))
                          (Functor.prod (tensor C) (𝟭 C) ⋙
                            tensor C ⋙
                              @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _))) :
                  (a : (F ⊠ G) ⊠ H ⟶ (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)
                (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
                      (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
                      (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) :
                    (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ (F ⊛ G) ⊛ H) →
                      (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (F ⊛ G) ⊛ H)
                  ((CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
                      (a : (F ⊛ G) ⊛ H ⟶ (F ⊛ G) ⊛ H) → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ (F ⊛ G) ⊛ H)
                    (𝟙 (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _))))))
            (x, y, z) ≫
          (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).map (Iso.hom (α_ x y z)) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z)) ⟶
        (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
dsimp :
  Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z)) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
    (Iso.inv (α_ (F.obj x) (G.obj y) (H.obj z)) ≫
      NatTrans.app
          ((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 (@convolution _ _ _ _ inst_2 inst_3 F 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)
            (Functor.whiskerLeft (prod.inverseAssociator C C C)
              (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
                    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
                    (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) :
                  (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ (F ⊛ G) ⊛ H) →
                    (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (F ⊛ G) ⊛ H)
                ((CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
                    (a : (F ⊛ G) ⊛ H ⟶ (F ⊛ G) ⊛ H) → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ (F ⊛ G) ⊛ H)
                  (𝟙 (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _))))))
          (x, y, z) ≫
        (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).map (Iso.hom (α_ x y z)))
    (Iso.inv (α_ (F.obj x) (G.obj y) (H.obj 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 :
          tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
            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 (@convolution _ _ _ _ inst_2 inst_3 F G _) H) (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z) ≫
            (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).map
              (Iso.hom (α_ x y z)) :
          tensorObj.{v₂, u₂} (𝒞 := inst_1)
              ((@convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y)) (H.obj z) ⟶
            (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z))))
⊢ Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z)) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
    (Iso.inv (α_ (F.obj x) (G.obj y) (H.obj z)) ≫
      NatTrans.app
          ((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 (@convolution _ _ _ _ inst_2 inst_3 F 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)
            (((FullyFaithful.homEquiv
                    (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.associativity C C C))) :
                  ((F ⊠ G) ⊠ H ⟶
                      Functor.prod (tensor C) (𝟭 C) ⋙
                        tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) ≃
                    (obj (obj (whiskeringLeft (C × C × C) ((C × C) × C) V) (prod.inverseAssociator C C C))
                        ((F ⊠ G) ⊠ H) ⟶
                      obj (obj (whiskeringLeft (C × C × C) ((C × C) × C) V) (prod.inverseAssociator C C C))
                        (Functor.prod (tensor C) (𝟭 C) ⋙
                          tensor C ⋙
                            @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _))) :
                (a : (F ⊠ G) ⊠ H ⟶ (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)
              (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
                    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
                    (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) :
                  (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ (F ⊛ G) ⊛ H) →
                    (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (F ⊛ G) ⊛ H)
                ((CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
                    (a : (F ⊛ G) ⊛ H ⟶ (F ⊛ G) ⊛ H) → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ (F ⊛ G) ⊛ H)
                  (𝟙 (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _))))))
          (x, y, z) ≫
        (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).map (Iso.hom (α_ x y z)))
    (Iso.inv (α_ (F.obj x) (G.obj y) (H.obj 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 :
          tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
            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 (@convolution _ _ _ _ inst_2 inst_3 F G _) H) (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z) ≫
            (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).map
              (Iso.hom (α_ x y z)) :
          tensorObj.{v₂, u₂} (𝒞 := inst_1)
              ((@convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y)) (H.obj z) ⟶
            (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z))))

Tactic:
dsimp [Functor.FullyFaithful.homEquiv, Equivalence.fullyFaithfulFunctor, prod.associativity]

Premises used: CategoryTheory.Equivalence.fullyFaithfulFunctor, CategoryTheory.Functor.FullyFaithful.homEquiv, CategoryTheory.prod.associativity