I am trying to understand definition (characterization) of projective module via dual basis lemma, compared with free module.
Let $F$ be a left $R$-module.
$F$ is said to be free if
(1) there is a generating set $(u_i)$ of $F$;
(2) there is a family $f_i$ of $R$-homomorphisms from $F$ to $R$ with following property:
(2.1) Every $x\in F$ can be written as $x=\sum_i f_i(x)u_i$, where all except finitely many $f_i(x)$ are $0$;
(2.2) The expression in $(2.1)$ is unique.
Q. The characterization of projective $R$-module via dual basis lemma can be obtained by dropping condition (2.2) in above definition. Is this assertion correct?
I mean $F$ is projective if and only if $(1)+(2)+(2.1)$ holds.