-- 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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 : DayConvolution F U
x : C
dsimp :
  Eq.{v₂ + 1} (α := F.obj x ⟶ (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).obj x)
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
      (Iso.inv (ρ_ (F.obj x)))
      (CategoryStruct.comp (Y :=
        (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (NatTrans.app
          ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (𝟙
                  (prod.rightInverseUnitor C ⋙ F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
                    prod.rightInverseUnitor C ⋙
                      Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙
                        tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)) :
              (a :
                  prod.rightInverseUnitor C ⋙ F ⊠ fromPUnit (𝟙_ V) ⟶
                    prod.rightInverseUnitor C ⋙ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C ⋙ F ⊛ U) →
                prod.rightInverseUnitor C ⋙ F ⊠ fromPUnit (𝟙_ V) ⟶
                  (prod.rightInverseUnitor C ⋙ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C) ⋙ F ⊛ U)
            (((FullyFaithful.homEquiv
                    (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.rightUnitorEquivalence C))) :
                  (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
                      Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙
                        tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _) ≃
                    (obj (obj (whiskeringLeft C (C × Discrete PUnit.{1}) V) (prod.rightInverseUnitor C))
                        (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) ⟶
                      obj (obj (whiskeringLeft C (C × Discrete PUnit.{1}) V) (prod.rightInverseUnitor C))
                        (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙
                          tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _))) :
                (a : F ⊠ fromPUnit (𝟙_ V) ⟶ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C ⋙ F ⊛ U) →
                  prod.rightInverseUnitor C ⋙ F ⊠ fromPUnit (𝟙_ V) ⟶
                    prod.rightInverseUnitor C ⋙ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C ⋙ F ⊛ U)
              (((F ⊠ U).homEquivOfIsLeftKanExtension (extensionUnitRight U (φ U) F)
                    (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _) :
                  (a : F ⊠ U ⟶ tensor C ⋙ F ⊛ U) →
                    F ⊠ fromPUnit (𝟙_ V) ⟶ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C ⋙ F ⊛ U)
                (((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).homEquivOfIsLeftKanExtension
                      (DayConvolution.unit F U) (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _) :
                    (a : F ⊛ U ⟶ F ⊛ U) → F ⊠ U ⟶ tensor C ⋙ F ⊛ U)
                  (𝟙 (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _))))))
          x)
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).map (Iso.hom (ρ_ x)))))
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
      (Iso.inv (ρ_ (F.obj x)))
      (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (F.obj x ◁ can)
        (CategoryStruct.comp (Y :=
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
          (NatTrans.app (DayConvolution.unit F U) (x, tensorUnit.{v₁, u₁} C (𝒞 := inst)))
          ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).map (Iso.hom (ρ_ x))))))
⊢ Eq (α := F.obj x ⟶ (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).obj x)
    (NatTrans.app (Iso.inv (rightUnitor U F)) x)
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
      (Iso.inv (ρ_ (F.obj x)))
      (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) ((F, U).2.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (F.obj x ◁ can)
        (CategoryStruct.comp (Y :=
          (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).obj
            (x, tensorUnit.{v₁, u₁} C (𝒞 := inst)))
          (NatTrans.app (DayConvolution.unit F U) (x, tensorUnit.{v₁, u₁} C (𝒞 := inst)))
          ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).map (Iso.hom (ρ_ x))))))

Tactic:
dsimp [rightUnitor, Coyoneda.fullyFaithful, corepresentableByRight,
    rightUnitorCorepresentingIso, Functor.CorepresentableBy.ofIso,
    Functor.corepresentableByEquiv, Iso.toEquiv, Equiv.toIso]

Premises used: CategoryTheory.Functor.corepresentableByEquiv, CategoryTheory.Iso.toEquiv, CategoryTheory.Coyoneda.fullyFaithful, Equiv.toIso, CategoryTheory.MonoidalCategory.DayConvolutionUnit.rightUnitor, CategoryTheory.MonoidalCategory.DayConvolutionUnit.corepresentableByRight, CategoryTheory.Functor.CorepresentableBy.ofIso, CategoryTheory.MonoidalCategory.DayConvolutionUnit.rightUnitorCorepresentingIso