Free module definition

Question

Let $X$ be non empty set and let $F$ be a left $R$ module. Left $R$ module with function $f:X\rightarrow A$ is called free on $X$ if exists unique homomorphism of $R$ modules $g: F\rightarrow A$ such that $g\circ l=f$.

Is it correct definition?

Answer 1

Given a (possibly empty) set $X$, a $R$-module $M$ and a function $f:X\to M$, the pair $(M,f)$ is said to be a free module over $X$ if for all $R$-modules $A$ and for all functions $g:X\to A$ there is exactly one $R$-module homomorphism $g':M\to A$ such that $g=g'\circ f$.