Lifting property of a free module ($\varphi q=\psi$)

For a free module $F$, a surjective module homomorphism $\varphi :P\to M$, and a module homomorphism $\psi :F\to M$, there is a module homomorphism $q:F\to P$ satisfying $\varphi q=\psi$. Choose a preimage under $\varphi$ of the image of each element of a basis of a module of $F$, and extend by linearity. Thus every free module is a projective module.