-- Module: Mathlib.Algebra.Category.MonCat.Limits
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))
refine_1 :
  ∀ (s : Cone F),
    EquivLike.coe (E :=
        (↑(F ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                    carrier)).sections :
            Type (max u v)) ≃
          Shrink.{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)))
        (equivShrink
          (↑(F ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                    carrier)).sections :
            Type (max u v)))
        (Subtype.mk (p :=
          fun
            (x :
              (j : J) →
                (F ⋙
                      forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                        carrier)).obj
                  j) ↦
          x ∈
            (F ⋙
                forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                  carrier)).sections)
          (fun (j : J) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app
                    (Cone.π
                      (Functor.mapCone
                        (forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                          carrier))
                        s))
                    j) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj ((forget MonCat).mapCone s).pt).obj j)) →
                  (fun (X : Type u) ↦ X) ((F ⋙ forget MonCat).obj j))
              1)
          ⋯) =
      1
refine_2 :
  ∀ (s : Cone F) (x y : (↑s.1 : Type u)),
    EquivLike.coe (E :=
        (↑(F ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                    carrier)).sections :
            Type (max u v)) ≃
          Shrink.{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)))
        (equivShrink
          (↑(F ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                    carrier)).sections :
            Type (max u v)))
        (Subtype.mk (p :=
          fun
            (x :
              (j : J) →
                (F ⋙
                      forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                        carrier)).obj
                  j) ↦
          x ∈
            (F ⋙
                forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                  carrier)).sections)
          (fun (j : J) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app
                    (Cone.π
                      (Functor.mapCone
                        (forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                          carrier))
                        s))
                    j) :
                (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj ((forget MonCat).mapCone s).pt).obj j)) →
                  (fun (X : Type u) ↦ X) ((F ⋙ forget MonCat).obj j))
              (x * y))
          ⋯) =
      EquivLike.coe (E :=
          (↑(F ⋙
                    forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                      carrier)).sections :
              Type (max u v)) ≃
            Shrink.{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)))
          (equivShrink
            (↑(F ⋙
                    forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                      carrier)).sections :
              Type (max u v)))
          (Subtype.mk (p :=
            fun
              (x :
                (j : J) →
                  (F ⋙
                        forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                          carrier)).obj
                    j) ↦
            x ∈
              (F ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                    carrier)).sections)
            (fun (j : J) ↦
              (ConcreteCategory.hom (FC := TypeCat.Fun)
                    (NatTrans.app
                      (Cone.π
                        (Functor.mapCone
                          (forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                            carrier))
                          s))
                      j) :
                  (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj ((forget MonCat).mapCone s).pt).obj j)) →
                    (fun (X : Type u) ↦ X) ((F ⋙ forget MonCat).obj j))
                x)
            ⋯) *
        EquivLike.coe (E :=
          (↑(F ⋙
                    forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                      carrier)).sections :
              Type (max u v)) ≃
            Shrink.{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)))
          (equivShrink
            (↑(F ⋙
                    forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                      carrier)).sections :
              Type (max u v)))
          (Subtype.mk (p :=
            fun
              (x :
                (j : J) →
                  (F ⋙
                        forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                          carrier)).obj
                    j) ↦
            x ∈
              (F ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                    carrier)).sections)
            (fun (j : J) ↦
              (ConcreteCategory.hom (FC := TypeCat.Fun)
                    (NatTrans.app
                      (Cone.π
                        (Functor.mapCone
                          (forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                            carrier))
                          s))
                      j) :
                  (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj ((forget MonCat).mapCone s).pt).obj j)) →
                    (fun (X : Type u) ↦ X) ((F ⋙ forget MonCat).obj j))
                y)
            ⋯)
⊢ IsLimit (limitCone F)

Tactic:
refine IsLimit.ofFaithful (forget MonCat) (Types.Small.limitConeIsLimit.{v, u} _)
    (fun s => ofHom { toFun := _, map_one' := ?_, map_mul' := ?_ }) (fun s => rfl)