-- 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) (tensorRight v)
inst_6 : DayConvolution U F
x : C
dsimp :
  Eq.{v₂ + 1} (α := F.obj x ⟶ (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _).obj x)
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) (F.obj x))
      (Iso.inv (λ_ (F.obj x)))
      (CategoryStruct.comp (Y :=
        (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
        (NatTrans.app
          ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (𝟙
                  (prod.leftInverseUnitor C ⋙ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ F ⟶
                    prod.leftInverseUnitor C ⋙
                      Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙
                        tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)) :
              (a :
                  prod.leftInverseUnitor C ⋙ fromPUnit (𝟙_ V) ⊠ F ⟶
                    prod.leftInverseUnitor C ⋙ (fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ U ⊛ F) →
                prod.leftInverseUnitor C ⋙ fromPUnit (𝟙_ V) ⊠ F ⟶
                  (prod.leftInverseUnitor C ⋙ (fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C) ⋙ U ⊛ F)
            (((FullyFaithful.homEquiv
                    (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.leftUnitorEquivalence C))) :
                  (fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ F ⟶
                      Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙
                        tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _) ≃
                    (obj (obj (whiskeringLeft C (Discrete PUnit.{1} × C) V) (prod.leftInverseUnitor C))
                        (fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ F) ⟶
                      obj (obj (whiskeringLeft C (Discrete PUnit.{1} × C) V) (prod.leftInverseUnitor C))
                        (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙
                          tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _))) :
                (a : fromPUnit (𝟙_ V) ⊠ F ⟶ (fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ U ⊛ F) →
                  prod.leftInverseUnitor C ⋙ fromPUnit (𝟙_ V) ⊠ F ⟶
                    prod.leftInverseUnitor C ⋙ (fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ U ⊛ F)
              (((U ⊠ F).homEquivOfIsLeftKanExtension (extensionUnitLeft U (φ U) F)
                    (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _) :
                  (a : U ⊠ F ⟶ tensor C ⋙ U ⊛ F) →
                    fromPUnit (𝟙_ V) ⊠ F ⟶ (fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ U ⊛ F)
                (((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _).homEquivOfIsLeftKanExtension
                      (DayConvolution.unit U F) (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _) :
                    (a : U ⊛ F ⟶ U ⊛ F) → U ⊠ F ⟶ tensor C ⋙ U ⊛ F)
                  (𝟙 (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _))))))
          x)
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _).map (Iso.hom (λ_ x)))))
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) (F.obj x))
      (Iso.inv (λ_ (F.obj x)))
      (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (F.obj x))
        (can ▷ F.obj x)
        (CategoryStruct.comp (Y :=
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
          (NatTrans.app (DayConvolution.unit U F) (tensorUnit.{v₁, u₁} C (𝒞 := inst), x))
          ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _).map (Iso.hom (λ_ x))))))
⊢ Eq (α := F.obj x ⟶ (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _).obj x)
    (NatTrans.app (Iso.inv (leftUnitor U F)) x)
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) (F.obj x))
      (Iso.inv (λ_ (F.obj x)))
      (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1) ((U, F).1.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (F.obj x)) (can ▷ F.obj x)
        (CategoryStruct.comp (Y :=
          (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _).obj
            (tensorUnit.{v₁, u₁} C (𝒞 := inst), x))
          (NatTrans.app (DayConvolution.unit U F) (tensorUnit.{v₁, u₁} C (𝒞 := inst), x))
          ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _).map (Iso.hom (λ_ x))))))

Tactic:
dsimp [leftUnitor, Coyoneda.fullyFaithful, corepresentableByLeft,
    leftUnitorCorepresentingIso, Functor.CorepresentableBy.ofIso,
    Functor.corepresentableByEquiv]

Premises used: CategoryTheory.Functor.corepresentableByEquiv, CategoryTheory.MonoidalCategory.DayConvolutionUnit.corepresentableByLeft, CategoryTheory.Coyoneda.fullyFaithful, CategoryTheory.MonoidalCategory.DayConvolutionUnit.leftUnitor, CategoryTheory.MonoidalCategory.DayConvolutionUnit.leftUnitorCorepresentingIso, CategoryTheory.Functor.CorepresentableBy.ofIso