Universal property of a quotient module

If an R-module homomorphism $f:M\to P$ vanishes on a submodule $N$, there is a unique homomorphism $\bar{f}:M/N\to P$ satisfying $f=\bar{f}\circ q$, where $q:M\to M/N$ is the quotient map. It is given by $\bar{f}(m+N)=f(m)$.