-- 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))
⊢ LiftsToLimit F (forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)) c t
have h : ∀ (x : J),
  Eq (α :=
    Functor.obj.{w, u, v, u + 1}
        (Functor.obj (Functor.const J)
          (Cone.pt (F :=
            F ⋙ forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier))
            (Functor.mapCone
              (forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier))
              {
                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 })))
        x ⟶
      (F ⋙ forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)).obj x)
    (NatTrans.app
      (Cone.π
        (Functor.mapCone
          (forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier))
          {
            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 } }))
      x)
    (Iso.hom
        (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)))))) ≫
      c.π.app x)
Tactic: refine LiftsToLimit.mk (LiftableCone.mk
    { pt := MonCat.of (Types.Small.limitCone (F ⋙ forget MonCat)).pt,
      π := NatTrans.mk
        (fun j => ofHom (limitπMonoidHom F j))
        (MonCat.HasLimits.limitCone F).π.naturality }
    (Cone.ext
      ((Types.isLimitEquivSections t).trans (equivShrink _)).symm.toIso
      (fun _ ↦ ?_))) ?_