Let $R$ be a commutative ring. If necessary, assume that $R$ has any convenient properties you like.

• Is there some $R$-module $Q$ such that an $R$-module $P$ is projective if and only if $\hom_R(P,Q) \to \hom_R(P,Q')$ is surjective for all quotients $Q'$ of $Q$?
• If yes, can $Q$ be chosen as a cogenerator?
• What happens when we restrict to finitely generated $P$?

Neither $Q=R$ nor $Q=\hom_{\mathbb{Z}}(R,\mathbb{Q}/\mathbb{Z})$ work. This question is a follow-up of math.SE/325495.