-- Lean 4 proof step from Mathlib.Algebra.Category.MonCat.Limits (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : CategoryTheory.Functor.{w, u, v, u + 1} J MonCat
inst_1 :
  Small.{u, max u v}
    (↑(F ⋙ forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)).sections :
      Type (max u v))
c : Cone (F ⋙ forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier))
t : IsLimit c
this :
  Small.{u, max u v}
    (↑(F ⋙ forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)).sections :
      Type (max u v))
⊢ IsLimit
    (LiftableCone.liftedCone (D := Type u) (F :=
      forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)) (c := c)
      {
        liftedCone :=
          {
            pt :=
              of
                (Types.Small.limitCone
                    (F ⋙
                      forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                        carrier))).pt,
            π :=
              { app := fun (j : J) ↦ ofHom (limitπMonoidHom F j), naturality := ⋯ :
                NatTrans
                  (Functor.obj (Functor.const J)
                    (of
                      (Types.Small.limitCone
                          (F ⋙
                            forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                              carrier))).pt))
                  F } :
            Cone F },
        validLift :=
          Cone.ext
            (Equiv.toIso
              (Equiv.symm
                ((Types.isLimitEquivSections t).trans
                  (equivShrink
                    (↑(F ⋙
                            forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                              carrier)).sections :
                      Type (max u v))))))
            ⋯ :
        LiftableCone F
          (forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)) c })
have h : (s : Cone F) →
  s.pt ⟶
    (LiftableCone.liftedCone (D := Type u) (F :=
        forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)) (c := c)
        {
          liftedCone :=
            {
              pt :=
                of
                  (Types.Small.limitCone
                      (F ⋙
                        forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                          carrier))).pt,
              π :=
                { app := fun (j : J) ↦ ofHom (limitπMonoidHom F j), naturality := sorry :
                  NatTrans
                    (Functor.obj (Functor.const J)
                      (of
                        (Types.Small.limitCone
                            (F ⋙
                              forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                                carrier))).pt))
                    F } :
              Cone F },
          validLift :=
            Cone.ext
              (Equiv.toIso
                (Equiv.symm
                  ((Types.isLimitEquivSections t).trans
                    (equivShrink
                      (↑(F ⋙
                              forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                                carrier)).sections :
                        Type (max u v))))))
              sorry :
          LiftableCone F
            (forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)) c }).pt
Tactic: refine IsLimit.ofFaithful (forget MonCat.{u}) (Types.Small.limitConeIsLimit.{v, u} _) ?_ ?_