Free module universal property

Free module universal property: Let $X$ be a set and let $F$ be a free $R$-module on $X$, equipped with a function $\eta:X\to F$. For any $R$-module $M$ and any function $g:X\to M$, there exists a unique $R$-module homomorphism $\tilde g:F\to M$ such that $\tilde g\circ \eta=g$.

This universal property characterizes free modules as the “linearizations” of sets , turning functions into module homomorphisms by unique extension.