Conditions of cantorness and co-cantorness for acts over the trivial semigroup

We find the necessary and sufficient conditions for an act $X$ over a trivial semigroup $\{e\}$ to be cantorian (or co-cantorian), i.e., for any act $Y$ the existence of injective (resp., surjective) homomorphisms $X \to Y$ and $Y \to X$ implies the isomorphism $X \cong Y$.