Show that the syntactic semigroup of $X$ is the smallest semigroup recognizing $X$ in the sense that, for every semigroup $S$ recognizing $X$, there exists a morphism from $S$ onto the syntactic semigroup of $X$.
Some background knowledge:
A semigroup $S$ is said to recognize a set $X$ if there exists a morphism $f:A^*\to S $ such that $X=f^{-1}(f(X))$.
(1) Let $X$ be a set of words. The set of contexts of a word $w$ is the set $C(w)=\{(x,y)\in A^*\times A^*| xwy\in X\}\;.$ (2) The syntactic equivalence of $X$ is defined by $u\equiv v$ if and only if $C(u)=C(v)$.
(3) The syntactic equivalence $\equiv$ is a congruence with concatenation of words and the quotient $A^*/\equiv$ is called the syntactic semigroup of $X$.
Actually I have known there exists a surjective map from $S$ to $A^*/\equiv$, since $\ker f\subset \equiv$. But I can't make sure the surjective map is a morphism.