-- 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
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : MonoidalCategoryStruct.{v₃, u₃} D
inst_6 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d d' d'' : D
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_8 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
a :
  Eq (α :=
    Functor.obj (C := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙
        Functor.obj (Functor.whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)) ⋙
          Functor.obj coyoneda
            (Opposite.op
              ((Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d ⊠
                  Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') ⊠
                Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))))
    ((Functor.CorepresentableBy.homEquiv
            (corepresentableBy₂' (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')) :
          (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊛ (ι C V D).obj d'' ⟶ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
            ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  (Functor.whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                    coyoneda.obj (Opposite.op (((ι C V D).obj d ⊠ (ι C V D).obj d') ⊠ (ι C V D).obj d''))).obj
              ((ι C V D).obj (d ⊗ d' ⊗ d'')))
        (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (α_ d d' d''))) :
      Functor.obj (C := Functor.{v₁, v₂, u₁, u₂} C V)
        (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙
          Functor.obj (Functor.whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)) ⋙
            Functor.obj coyoneda
              (Opposite.op
                ((Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d ⊠
                    Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') ⊠
                  Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')))
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))))
    ((Functor.CorepresentableBy.homEquiv
            (corepresentableBy₂' (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')) :
          (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊛ (ι C V D).obj d'' ⟶ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
            ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  (Functor.whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                    coyoneda.obj (Opposite.op (((ι C V D).obj d ⊠ (ι C V D).obj d') ⊠ (ι C V D).obj d''))).obj
              ((ι C V D).obj (d ⊗ d' ⊗ d'')))
        (Iso.hom
          (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d''))) :
      Functor.obj (C := Functor.{v₁, v₂, u₁, u₂} C V)
        (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙
          Functor.obj (Functor.whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)) ⋙
            Functor.obj coyoneda
              (Opposite.op
                ((Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d ⊠
                    Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d') ⊠
                  Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')))
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))))
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_4) (tensorObj.{v₃, u₃} (𝒞 := inst_4) d d') d''))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))))
    (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (α_ d d' d'')))
    (Iso.hom
      (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d')
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')))

Tactic:
apply corepresentableBy₂'
    (ι C V D |>.obj d) (ι C V D |>.obj d') (ι C V D |>.obj d'') |>.homEquiv.injective

Premises used: CategoryTheory.Functor.CorepresentableBy.homEquiv, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy₂', Equiv.injective, CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.ι, CategoryTheory.Functor.obj