-- 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
this :
  Eq.{v₂ + 1} (α :=
    (Opposite.unop (Opposite.op (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))))).obj
        (x, { as := PUnit.unit : Discrete PUnit.{1} }) ⟶
      (obj
            (obj (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V)
              (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))))
            (obj (obj (whiskeringLeft (C × C) C V) (tensor C)) F)).obj
        (x, { as := PUnit.unit }))
    ((fun
          (t :
            (obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
                  obj (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V)
                      (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) ⋙
                    obj coyoneda (Opposite.op (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))))).obj
              F) ↦
        (t.app (x, { as := PUnit.unit }) :
          (Opposite.unop (Opposite.op (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))))).obj
              (x, { as := PUnit.unit : Discrete PUnit.{1} }) ⟶
            (obj
                  (obj (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V)
                    (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))))
                  (obj (obj (whiskeringLeft (C × C) C V) (tensor C)) F)).obj
              (x, { as := PUnit.unit })))
      ((CorepresentableBy.homEquiv (corepresentableByRight U F) :
          (a : F ⊛ U ⟶ F) →
            ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                  (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj ((𝟭 C).prod (fromPUnit (𝟙_ C))) ⋙
                    coyoneda.obj (Opposite.op (F ⊠ fromPUnit (𝟙_ V)))).obj
              F)
        ((Equiv.symm (CorepresentableBy.homEquiv (corepresentableByRight U F)) :
            (a :
                ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                      (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj ((𝟭 C).prod (fromPUnit (𝟙_ C))) ⋙
                        coyoneda.obj (Opposite.op (F ⊠ fromPUnit (𝟙_ V)))).obj
                  F) →
              F ⊛ U ⟶ F)
          ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app (Iso.hom (Iso.symm (rightUnitorCorepresentingIso F))) F) :
              (a : (fun (X : Type (max u₁ v₂)) ↦ X) ((coyoneda.obj (Opposite.op F)).obj F)) →
                (fun (X : Type (max u₁ v₂)) ↦ X)
                  (((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                        (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj ((𝟭 C).prod (fromPUnit (𝟙_ C))) ⋙
                          coyoneda.obj (Opposite.op (F ⊠ fromPUnit (𝟙_ V)))).obj
                    F))
            (𝟙 (Opposite.unop (Opposite.op F)))))))
    ((fun
          (t :
            (obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
                  obj (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V)
                      (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) ⋙
                    obj coyoneda (Opposite.op (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))))).obj
              F) ↦
        (t.app (x, { as := PUnit.unit }) :
          (Opposite.unop (Opposite.op (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))))).obj
              (x, { as := PUnit.unit : Discrete PUnit.{1} }) ⟶
            (obj
                  (obj (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V)
                    (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))))
                  (obj (obj (whiskeringLeft (C × C) C V) (tensor C)) F)).obj
              (x, { as := PUnit.unit })))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app (Iso.hom (Iso.symm (rightUnitorCorepresentingIso F))) F) :
          (a : (fun (X : Type (max u₁ v₂)) ↦ X) ((coyoneda.obj (Opposite.op F)).obj F)) →
            (fun (X : Type (max u₁ v₂)) ↦ X)
              (((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                    (whiskeringLeft (C × Discrete PUnit.{1}) (C × C) V).obj ((𝟭 C).prod (fromPUnit (𝟙_ C))) ⋙
                      coyoneda.obj (Opposite.op (F ⊠ fromPUnit (𝟙_ V)))).obj
                F))
        (𝟙 (Opposite.unop (Opposite.op F)))))
dsimp :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
      (F.obj x ◁ can)
      (NatTrans.app (DayConvolution.unit F U) (x, tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).descOfIsLeftKanExtension (DayConvolution.unit F U) F
              ((F ⊠ U).descOfIsLeftKanExtension (L :=
                Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (F :=
                F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) (extensionUnitRight U (φ U) F) (tensor C ⋙ F)
                (((Equiv.symm
                        (FullyFaithful.homEquiv
                          (Equivalence.fullyFaithfulFunctor (Equivalence.congrLeft (prod.rightUnitorEquivalence C)))) :
                      (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 ⋙ F) ≃
                        (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
                          Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙ tensor C ⋙ F)) :
                    (a :
                        prod.rightInverseUnitor C ⋙ F ⊠ fromPUnit (𝟙_ V) ⟶
                          prod.rightInverseUnitor C ⋙ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C ⋙ F) →
                      F ⊠ fromPUnit (𝟙_ V) ⟶ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C ⋙ F)
                  ((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 ⋙ F)) :
                      (a :
                          prod.rightInverseUnitor C ⋙ F ⊠ fromPUnit (𝟙_ V) ⟶
                            (prod.rightInverseUnitor C ⋙ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C) ⋙ F) →
                        prod.rightInverseUnitor C ⋙ F ⊠ fromPUnit (𝟙_ V) ⟶
                          prod.rightInverseUnitor C ⋙ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C ⋙ F)
                    ((Iso.inv (NatIso.ofComponents (fun (x : C) ↦ Iso.symm (ρ_ (F.obj x))) ⋯) ≫ 𝟙 F) ≫
                      NatTrans.app (map (whiskeringLeft C C V) (Iso.inv (NatIso.ofComponents (fun (x : C) ↦ ρ_ x) ⋯)))
                        F))))).app
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst)))))
    (Iso.hom (ρ_ (F.obj x)) ≫ F.map (Iso.inv (ρ_ x)))
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
      F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (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)))
        (NatTrans.app (Iso.hom (rightUnitor U F))
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))))
    (Iso.hom (ρ_ (F.obj x)) ≫ F.map (Iso.inv (ρ_ x)))

Tactic:
dsimp [rightUnitor, Coyoneda.fullyFaithful, corepresentableByRight,
    rightUnitorCorepresentingIso, Functor.CorepresentableBy.ofIso,
    Functor.corepresentableByEquiv] at this ⊢

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