-- 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 : InducedLawfulDayConvolutionMonoidalCategoryStructCore.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorRight v)
inst_9 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (MonoidalCategoryStruct.tensorUnit.{v₁, u₁} C (𝒞 := inst))) d)
      (tensorLeft v)
d₁ d₁' d₂ 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 _ _) d₁ ⊠ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁')))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
        (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂')))
    ((Functor.CorepresentableBy.homEquiv
            (DayConvolution.corepresentableBy (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁)
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁')) :
          (a : (ι C V D).obj d₁ ⊛ (ι C V D).obj d₁' ⟶ (ι C V D).obj (tensorObj C V 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 (tensorObj C V d₂ d₂')))
        (Iso.hom (tensorObjIsoConvolution C V d₁ d₁') ≫
          DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
              (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f') ≫
            Iso.inv (tensorObjIsoConvolution C V d₂ d₂')) :
      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 _ _) d₁ ⊠ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁')))
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
          (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂')))
    ((Functor.CorepresentableBy.homEquiv
            (DayConvolution.corepresentableBy (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁)
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁')) :
          (a : (ι C V D).obj d₁ ⊛ (ι C V D).obj d₁' ⟶ (ι C V D).obj (tensorObj C V 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 (tensorObj C V d₂ d₂')))
        (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
          (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) 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 _ _) d₁ ⊠ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _) d₁')))
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
          (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ 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 _ _)
        (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₁ d₁'))
      (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ _)
        (@tensorObj.{v₁, v₂, v₃, u₁, u₂, u₃} C inst V inst_1 inst_2 inst_3 _ inst_4 _ d₂ d₂')))
    (Iso.hom (tensorObjIsoConvolution C V d₁ d₁') ≫
      DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
          (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f') ≫
        Iso.inv (tensorObjIsoConvolution C V d₂ d₂'))
    (DayConvolution.map (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f)
      (Functor.map (@ι C _ V _ inst_2 inst_3 D _ _) f'))

Tactic:
apply (convolutions C V d₁ d₁').corepresentableBy.homEquiv.injective

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