-- 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.{v₂ + 1} (α :=
    (Opposite.unop (Opposite.op ((F ⊠ G) ⊠ H))).obj ((x, y), z) ⟶
      (obj (obj (whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)))
            (obj (obj (whiskeringLeft (C × C) C V) (tensor C))
              (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _))).obj
        ((x, y), z))
    ((fun
          (t :
            (obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
                  obj (whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)) ⋙
                    obj coyoneda (Opposite.op ((F ⊠ G) ⊠ H))).obj
              (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) ↦
        (t.app ((x, y), z) :
          (Opposite.unop (Opposite.op ((F ⊠ G) ⊠ H))).obj ((x, y), z) ⟶
            (obj (obj (whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)))
                  (obj (obj (whiskeringLeft (C × C) C V) (tensor C))
                    (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _))).obj
              ((x, y), z)))
      ((CorepresentableBy.homEquiv (corepresentableBy₂' F G H) :
          (a : (F ⊛ G) ⊛ H ⟶ F ⊛ G ⊛ H) →
            ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  (whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                    coyoneda.obj (Opposite.op ((F ⊠ G) ⊠ H))).obj
              (F ⊛ G ⊛ H))
        ((Equiv.symm (CorepresentableBy.homEquiv (corepresentableBy₂' F G H)) :
            (a :
                ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                      (whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                        coyoneda.obj (Opposite.op ((F ⊠ G) ⊠ H))).obj
                  (F ⊛ G ⊛ H)) →
              (F ⊛ G) ⊛ H ⟶ F ⊛ G ⊛ H)
          ((CorepresentableBy.homEquiv
                (CorepresentableBy.ofIso (corepresentableBy₂ F G H) (Iso.symm (associatorCorepresentingIso F G H))) :
              (a : F ⊛ G ⊛ H ⟶ F ⊛ G ⊛ H) →
                ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                      (whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                        coyoneda.obj (Opposite.op ((F ⊠ G) ⊠ H))).obj
                  (F ⊛ G ⊛ H))
            (𝟙 (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _))))))
    ((fun
          (t :
            (obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
                  obj (whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)) ⋙
                    obj coyoneda (Opposite.op ((F ⊠ G) ⊠ H))).obj
              (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)) ↦
        (t.app ((x, y), z) :
          (Opposite.unop (Opposite.op ((F ⊠ G) ⊠ H))).obj ((x, y), z) ⟶
            (obj (obj (whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)))
                  (obj (obj (whiskeringLeft (C × C) C V) (tensor C))
                    (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _))).obj
              ((x, y), z)))
      ((CorepresentableBy.homEquiv
            (CorepresentableBy.ofIso (corepresentableBy₂ F G H) (Iso.symm (associatorCorepresentingIso F G H))) :
          (a : F ⊛ G ⊛ H ⟶ F ⊛ G ⊛ H) →
            ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  (whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                    coyoneda.obj (Opposite.op ((F ⊠ G) ⊠ H))).obj
              (F ⊛ G ⊛ H))
        (𝟙 (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _))))
dsimp :
  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 :=
                      (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))
    (Iso.hom (α_ (F.obj x) (G.obj y) (H.obj 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) :
            tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z)) ⟶
              tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x)
                ((@convolution _ _ _ _ inst_2 inst_3 G H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
          (NatTrans.app (unit F (@convolution _ _ _ _ inst_2 inst_3 G H _)) (x, tensorObj.{v₁, u₁} (𝒞 := inst) y z) ≫
              (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).map
                (Iso.inv (α_ x y z)) :
            tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x)
                ((@convolution _ _ _ _ inst_2 inst_3 G H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) y 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)) :
      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) ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F G _).obj (x, y)) (H.obj z))
      (NatTrans.app (unit F G) (x, y) ▷ H.obj z)
      (CategoryStruct.comp (Y :=
        (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z))
        (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H) (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z))
        (NatTrans.app (Iso.hom (associator F G H))
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))) =
    Iso.hom (α_ (F.obj x) ((G, H).1.obj (y, z).1) ((G, H).2.obj (y, z).2)) ≫
      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) ≫
          (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).map
            (Iso.inv (α_ (x, tensorObj.{v₁, u₁} (𝒞 := inst) y z).1 y z))

Tactic:
dsimp [associator, Coyoneda.fullyFaithful, corepresentableBy₂,
    corepresentableBy₂', Functor.CorepresentableBy.ofIso, corepresentableBy₂,
    Functor.corepresentableByEquiv, associatorCorepresentingIso] at this ⊢

Premises used: CategoryTheory.Functor.corepresentableByEquiv, CategoryTheory.MonoidalCategory.DayConvolution.associatorCorepresentingIso, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy₂', CategoryTheory.Coyoneda.fullyFaithful, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy₂, CategoryTheory.MonoidalCategory.DayConvolution.associator, CategoryTheory.Functor.CorepresentableBy.ofIso