Universal property of a Verma module

If a $\mathfrak{g}$-module contains a vector $v$ of weight $\lambda$ annihilated by ${\mathfrak{n}}^{+}$, there is a unique homomorphism $M_{\lambda }\to V$ sending the canonical highest-weight vector to $v$. Every highest-weight module of weight $\lambda$ is therefore a quotient of $M_{\lambda }$.