-- 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 :=
    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)
                    (CategoryStruct.comp (Y := 𝟭 C ⋙ F)
                      (Iso.inv (NatIso.ofComponents (fun (x : C) ↦ Iso.symm (ρ_ (F.obj x))) ⋯))
                      (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)))))
    (NatTrans.app
      (((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)
          (CategoryStruct.comp (Y := 𝟭 C ⋙ F) (Iso.inv (NatIso.ofComponents (fun (x : C) ↦ Iso.symm (ρ_ (F.obj x))) ⋯))
            (NatTrans.app (map (whiskeringLeft C C V) (Iso.inv (NatIso.ofComponents (fun (x : C) ↦ ρ_ x) ⋯))) F))))
      (x, { as := PUnit.unit }))
simp :
  Eq.{v₂ + 1} (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))))
    (NatTrans.app
        (((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)
            (CategoryStruct.comp (Y := 𝟭 C ⋙ F)
              (Iso.inv (NatIso.ofComponents (fun (x : C) ↦ Iso.symm (ρ_ (F.obj x))) ⋯))
              (NatTrans.app (map (whiskeringLeft C C V) (Iso.inv (NatIso.ofComponents (fun (x : C) ↦ ρ_ x) ⋯))) F))))
        (x, { as := PUnit.unit }) :
      (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))).obj (x, { as := PUnit.unit : Discrete PUnit.{1} }) ⟶
        (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙ tensor C ⋙ F).obj
          (x, { as := PUnit.unit }))
    (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) (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)))

Tactic:
simp only [Category.comp_id, this]

Premises used: CategoryTheory.Category.comp_id