# Projective module vs. Free module: dual basis lemma

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.