Generated submagma of a free magma

Question

Let $X$ be a set and $S\subset X$. Let $M(X)$ denote the free magma constructed on $X$ and $i:S\hookrightarrow X $ be the canonical injection of $S$ into $X$. We know that there exists a unique injective morphism $$M(i):M(S)\hookrightarrow M(X)$$ which agrees with $i$ on $S$. Let $M'(S):=M(i)(M(S))$. Then the mapping $M(S)\rightarrow M'(S)$ is an isomorphism. Now, let $$\mathcal{X}=\{Y\subset M(X)\ |\ YY\subset Y \land S\subset Y\};$$ $\bigcap_{Y\in\mathcal{X}}Y$ is the submagma of $M(X)$ generated by $S$. Upon identification of $M(S)$ and $M'(S)$, is it true that $$\bigcap_{Y\in\mathcal{X}}Y=M(S)?$$

Answer 1

No, because $M'(S) - \{e\} \in \mathcal X$, where $e$ is the identity of $M(X)$.

Conversely, it is clear that every $Y \in \mathcal X$ must contain $M'(S) - \{e\}$, so that $$\bigcap_{Y \in \mathcal X} Y = M'(S) - \{e\}$$