Let $M$ be a generator for the category of left $R$-modules, and let we have an $R$-epimorphism $h$ from $R^{(X)}$ to an $R$-module $P$ which is projective relative to $M^{(X)}$ ($X$ is a set). I want to prove that $P$ is indeed projective.
I tried to find, for an arbitrary left $R$-module $N$, an $R$-epimorphism $f$ from $M^{(X)}$ to $N $ for, due to the short exact sequence $0→ker(f)→M^{(X)}→N→0$, if $P$ is $M^{(X)}$-projective then $P$ is projective relative to both $N$ and $ker(f)$. Since there is an $R$-epimorphism from a direct sum $M^{(Y)}$ to $N$, if $|X|$ is greater than or equal to $|Y|$ we are done because there is a projection from $M^{(X)}$ to $M^{(Y)}$. But, for the other case which may use the epimorphism $h$ I am stuck! Thanks for any help!