$\newcommand{\Tor}{\operatorname{Tor}}$In the category of R-modules, a morphism $f:M\to N$ is called a phantom morphism if for every finitely presented module $F$ and every morphism $g:F\to M$, $fg$ factors through a projective module, i.e there exist a projective module P and morphisms $k:F\to P$ and $h:P\to N$ such that $hk=fg$.
I want to show that $f$ is a phantom morphism if and only if the induced morphism of abelian groups $\Tor_{1}(F,f): \Tor_{1}(F,M) \to \Tor_{1}(F,N)$ is zero for every finitely presented module F.
I have proved that if $f$ is a phantom morphism then the induced morphism of abelian groups $\Tor_{1}(F,f): \Tor_{1}(F,M) \to \Tor_{1}(F,N)$ is zero for every finitely presented module $F$. But I have no idea about the converse. Any help?