Let $X$ be a compact Riemann surface of genus $g$. Consider the set $$M=\{(X,\{x_1,\dots,x_n\}): x_k\in X \text{ }\&\text{ } x_i\ne x_j\text{ for }i\ne j\}.$$ Define an equivalence relation on $M$ in the following way: $(X,\{x_1,\dots,x_n\}) \sim (X',\{x_1',\dots,x_n'\}) $ iff there exists a biholomorphic map $f: X\rightarrow X'$ of Riemann surfaces such that $f(x_i)=x_i'$ for every $i \in \{1,\dots,n\}$.
Is it correct to say that, by definition, $\mathcal{M}_{g,n}=M/\sim$ ?