-- 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
inst_7 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft 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 × Discrete PUnit.{1}) (C × C) V)
            (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) ⋙
          Functor.obj coyoneda
            (Opposite.op
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d ⊠
                Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
    ((Functor.CorepresentableBy.homEquiv
            (corepresentableByRight
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
              (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 (𝟙_ D) ⟶ (ι C V D).obj d) →
            ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  (Functor.whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj
                      ((𝟭 C).prod (Functor.fromPUnit (𝟙_ C))) ⋙
                    coyoneda.obj (Opposite.op ((ι C V D).obj d ⊠ Functor.fromPUnit (𝟙_ V)))).obj
              ((ι C V D).obj d))
        (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (ρ_ 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 × Discrete PUnit.{1}) (C × C) V)
              (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) ⋙
            Functor.obj coyoneda
              (Opposite.op
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d ⊠
                  Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))))
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
    ((Functor.CorepresentableBy.homEquiv
            (corepresentableByRight
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
              (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 (𝟙_ D) ⟶ (ι C V D).obj d) →
            ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  (Functor.whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj
                      ((𝟭 C).prod (Functor.fromPUnit (𝟙_ C))) ⋙
                    coyoneda.obj (Opposite.op ((ι C V D).obj d ⊠ Functor.fromPUnit (𝟙_ V)))).obj
              ((ι C V D).obj d))
        (Iso.hom
          (DayConvolutionUnit.rightUnitor
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (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 × Discrete PUnit.{1}) (C × C) V)
              (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) ⋙
            Functor.obj coyoneda
              (Opposite.op
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d ⊠
                  Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))))
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) 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) d (tensorUnit.{v₃, u₃} D (𝒞 := inst_4))))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
    (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (ρ_ d)))
    (Iso.hom
      (DayConvolutionUnit.rightUnitor
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)))

Tactic:
apply corepresentableByRight
    (ι C V D |>.obj <| 𝟙_ D) (ι C V D |>.obj d) |>.homEquiv.injective

Premises used: CategoryTheory.Functor.CorepresentableBy.homEquiv, CategoryTheory.MonoidalCategory.DayConvolutionUnit.corepresentableByRight, CategoryTheory.MonoidalCategoryStruct.tensorUnit, Equiv.injective, CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.ι, CategoryTheory.Functor.obj