-- 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))
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))
s : Cone F
x y : (↑s.1 : Type u)
h :
  (equivShrink
          (↑(F ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                    carrier)).sections :
            Type (max u v)) :
        (a : (↑(F ⋙ forget MonCat).sections : Type (max u v))) →
          Shrink.{u, max u v} (↑(F ⋙ forget MonCat).sections : Type (max u v)))
      (Subtype.mk (p :=
        fun
          (x :
            (j : J) →
              ToType (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier) (F.obj j)) ↦
        x ∈
          (F ⋙
              forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                carrier)).sections)
        (fun (j : J) ↦
          HMul.hMul (α := (↑(F.obj j) : Type u)) (β := (↑(F.obj j) : Type u))
            ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)
                    (s.π.app j) :
                  (a : (↑s.pt : Type u)) → (↑(F.obj j) : Type u))
                x :
              (↑(F.obj j) : Type u))
            ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)
                    (s.π.app j) :
                  (a : (↑s.pt : Type u)) → (↑(F.obj j) : Type u))
                y :
              (↑(F.obj j) : Type u)))
        ⋯) =
    HMul.hMul (α :=
      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)))
      (β :=
      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)) :
            (a : (↑(F ⋙ forget MonCat).sections : Type (max u v))) →
              Shrink.{u, max u v} (↑(F ⋙ forget MonCat).sections : Type (max u v)))
          (Subtype.mk (p :=
            fun
              (x :
                (j : J) →
                  ToType (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)
                    (F.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 := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)
                    (s.π.app j) :
                  (a : (↑s.pt : Type u)) → (↑(F.obj j) : Type u))
                x)
            ⋯) :
        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)) :
            (a : (↑(F ⋙ forget MonCat).sections : Type (max u v))) →
              Shrink.{u, max u v} (↑(F ⋙ forget MonCat).sections : Type (max u v)))
          (Subtype.mk (p :=
            fun
              (x :
                (j : J) →
                  ToType (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)
                    (F.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 := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier)
                    (s.π.app j) :
                  (a : (↑s.pt : Type u)) → (↑(F.obj j) : Type u))
                y)
            ⋯) :
        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)))
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun)
          (IsLimit.lift
            (Types.Small.limitConeIsLimit
              (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)) s)) :
        (a : (fun (X : Type u) ↦ X) ((forget MonCat).mapCone s).pt) →
          (fun (X : Type u) ↦ X) (Types.Small.limitCone (F ⋙ forget MonCat)).pt)
      (x * y) =
    HMul.hMul (α :=
      (Types.Small.limitCone
          (F ⋙ forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier))).pt)
      (β :=
      (Types.Small.limitCone
          (F ⋙ forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier))).pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (IsLimit.lift
                (Types.Small.limitConeIsLimit
                  (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))
                  s)) :
            (a : (fun (X : Type u) ↦ X) ((forget MonCat).mapCone s).pt) →
              (fun (X : Type u) ↦ X) (Types.Small.limitCone (F ⋙ forget MonCat)).pt)
          x :
        (fun (X : Type u) ↦ X)
          (Types.Small.limitCone
              (F ⋙
                forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier))).pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (IsLimit.lift
                (Types.Small.limitConeIsLimit
                  (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))
                  s)) :
            (a : (fun (X : Type u) ↦ X) ((forget MonCat).mapCone s).pt) →
              (fun (X : Type u) ↦ X) (Types.Small.limitCone (F ⋙ forget MonCat)).pt)
          y :
        (fun (X : Type u) ↦ X)
          (Types.Small.limitCone
              (F ⋙
                forget MonCat (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := carrier))).pt)

Tactic:
simp only [Types.Small.limitCone_pt, Functor.comp_obj, Functor.mapCone_pt,
        Types.Small.limitConeIsLimit_lift, Functor.const_obj_obj, Functor.mapCone_π_app,
        ConcreteCategory.hom_ofHom, TypeCat.Fun.coe_mk, map_mul]

Premises used: CategoryTheory.Functor.comp_obj, TypeCat.Fun.coe_mk, CategoryTheory.Limits.Types.Small.limitCone_pt, CategoryTheory.Functor.mapCone_pt, map_mul, CategoryTheory.Limits.Types.Small.limitConeIsLimit_lift, CategoryTheory.Functor.mapCone_π_app, CategoryTheory.Functor.const_obj_obj, CategoryTheory.ConcreteCategory.hom_ofHom