-- Lean 4 proof step from Mathlib.Order.Category.PartOrdEmb (goal)
Goal:
J : Type u
inst : SmallCategory J
inst_1 : IsFiltered J
F : J ⥤ PartOrdEmb
c :
  Cocone (F ⋙ forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier))
hc : IsColimit c
s : Cocone F
j : J
x' y' :
  (F ⋙ forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier)).obj j
hx :
  Eq (α := ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)
    (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
              (hc.desc
                (Functor.mapCocone
                  (forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                    carrier))
                  s)) :
            TypeCat.Fun c.pt
              (ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)) :
          (a : c.pt) → ToType s.pt)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) x' : c.pt) :
      ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)
    ((ConcreteCategory.hom (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier)
            (s.ι.app j) :
          (a : (↑(F.obj j) : Type u)) → (↑s.pt : Type u))
        x' :
      (↑s.pt : Type u))
hy :
  Eq (α := ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)
    (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
              (hc.desc
                (Functor.mapCocone
                  (forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                    carrier))
                  s)) :
            TypeCat.Fun c.pt
              (ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)) :
          (a : c.pt) → ToType s.pt)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) y' : c.pt) :
      ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)
    ((ConcreteCategory.hom (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier)
            (s.ι.app j) :
          (a : (↑(F.obj j) : Type u)) → (↑s.pt : Type u))
        y' :
      (↑s.pt : Type u))
⊢ x' ≤ y' ↔
    LE.le (α := CoconePt hc)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) x' : c.pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) y' : c.pt)
have refine : LE.le (α := CoconePt hc)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) x' : c.pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) y' : c.pt) →
  ∀ (k : J) (x y : (↑(F.obj k) : Type u))
    (hx' :
      Eq (α := (fun (X : Type u) ↦ X) (Functor.obj (Functor.obj (Functor.const J) c.pt) k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app k) :
              (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj k)) →
                (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj k))
            x :
          (fun (X : Type u) ↦ X) (Functor.obj (Functor.obj (Functor.const J) c.pt) k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) x' : c.pt))
    (hy' :
      Eq (α := (fun (X : Type u) ↦ X) (Functor.obj (Functor.obj (Functor.const J) c.pt) k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app k) :
              (a : (fun (X : Type u) ↦ X) ((F ⋙ forget PartOrdEmb).obj k)) →
                (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj k))
            y :
          (fun (X : Type u) ↦ X) (Functor.obj (Functor.obj (Functor.const J) c.pt) k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) y' : c.pt))
    (h : x ≤ y), x' ≤ y'
Tactic: refine ⟨fun h ↦ ⟨j, _, _, rfl, rfl, h⟩, fun ⟨k, x, y, hx', hy', h⟩ ↦ ?_⟩