-- 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
d₁ d₂ d₁' d₂' : D
f : d₁ ⟶ d₂
f' : d₁' ⟶ d₂'
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 coyoneda
          (Opposite.op
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁ ⊠
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁')))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorObj.{v₃, u₃} (𝒞 := inst_4) d₂ d₂')))
    ((Functor.CorepresentableBy.homEquiv
            (DayConvolution.corepresentableBy (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁)
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁')) :
          (a : (ι C V D).obj d₁ ⊛ (ι C V D).obj d₁' ⟶ (ι C V D).obj (d₂ ⊗ d₂')) →
            ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  coyoneda.obj (Opposite.op ((ι C V D).obj d₁ ⊠ (ι C V D).obj d₁'))).obj
              ((ι C V D).obj (d₂ ⊗ d₂')))
        (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (f ⊗ₘ f')) :
      Functor.obj (C := Functor.{v₁, v₂, u₁, u₂} C V)
        (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙
          Functor.obj coyoneda
            (Opposite.op
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁ ⊠
                Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁')))
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorObj.{v₃, u₃} (𝒞 := inst_4) d₂ d₂')))
    ((Functor.CorepresentableBy.homEquiv
            (DayConvolution.corepresentableBy (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁)
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁')) :
          (a : (ι C V D).obj d₁ ⊛ (ι C V D).obj d₁' ⟶ (ι C V D).obj (d₂ ⊗ d₂')) →
            ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  coyoneda.obj (Opposite.op ((ι C V D).obj d₁ ⊠ (ι C V D).obj d₁'))).obj
              ((ι C V D).obj (d₂ ⊗ d₂')))
        (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) f)
          (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) f')) :
      Functor.obj (C := Functor.{v₁, v₂, u₁, u₂} C V)
        (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙
          Functor.obj coyoneda
            (Opposite.op
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁ ⊠
                Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁')))
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorObj.{v₃, u₃} (𝒞 := inst_4) d₂ d₂')))
⊢ 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) d₁ d₁'))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorObj.{v₃, u₃} (𝒞 := inst_4) d₂ d₂')))
    (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (f ⊗ₘ f'))
    (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) f)
      (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) f'))

Tactic:
apply DayConvolution.corepresentableBy
    (ι C V D |>.obj d₁) (ι C V D |>.obj d₁') |>.homEquiv.injective

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