import Mathlib

open CategoryTheory

namespace CAT_statement_S_0014

universe u uX

variable {X : Type uX} [Category.{vX} X]

namespace AHS2

structure ConcreteCat (X : Type v) [Category X] where
  C : Type u
  [cat : Category C]
  U : C ⥤ X
  [U_Faithful : U.Faithful]

attribute [instance] ConcreteCat.cat ConcreteCat.U_Faithful


def IsInitialHom {C : ConcreteCat (X:= X)} {A B : C.C} (f : A ⟶ B) : Prop :=
  ∀ ⦃Z : C.C⦄ (g : C.U.obj Z ⟶ C.U.obj A),
    (∃ h : Z ⟶ B, C.U.map h = g ≫ C.U.map f) →
      (∃ k : Z ⟶ A, C.U.map k = g)


def IsEmbedding {C : ConcreteCat (X:= X)} {A B : C.C} (f : A ⟶ B) : Prop :=
  IsInitialHom f ∧ Mono (C.U.map f)


def IsInjectiveObj {C : ConcreteCat (X:= X)} (I : C.C) : Prop :=
  ∀ ⦃A B : C.C⦄ (m : A ⟶ B),
    IsEmbedding  m →
    ∀ (f : A ⟶ I), ∃ g : B ⟶ I, m ≫ g = f

end AHS2

namespace Poset
def PosetConcrete : AHS2.ConcreteCat (Type u) where
  C := PartOrd.{u}
  cat := inferInstance
  U := forget PartOrd
  U_Faithful := inferInstance


theorem injective_iff_suplattice (P : PartOrd.{u}) :
  AHS2.IsInjectiveObj (C := PosetConcrete) P ↔ ∀ (s : Set P), ∃ x, IsLUB s x := by
  sorry

end Poset

end CAT_statement_S_0014