Right homotopic maps iff chain homotopic Assume the model structure on $Ch(R)$ (chain complexes of left modules over the ring $R$) in which fibrations are dimensionwise epimorphisms (i.e. surjections) and weak equivalences are homology isomorphisms (I don't need cofibrations so I won't describe them).
As the title suggests, I need to prove that two chain maps $f,g:B \to X$ are right homotopic $\iff$ they're chain homotopic. The "$\Longleftarrow$" side is easy, since Hovey provides a path object that does the job ($P_n:=X_n\oplus X_n\oplus X_{n+1}$ with $d(x,y,z):=(dx,dy, -dz+x-y))$. 
How can I prove the "$\Longrightarrow$" side, given that the right homotopy could be realized with any path object, not necessarily the one suggested? By the way, I can show it if I assume that the homotopy is indeed obtained by using that particular path object.
Thanks in advance for any hint!!

I tried using the cofibrant replacement functor, since a cofibrant domain simplify the problem (as expressed in the comments).
What I've obtained is that $f \circ q_B \sim g \circ q_B$ (chain homotopic) where $q_B$ is a trivial fibration, since now any path object works and so I can use the one suggested by the author.But unfortunately it doesn't seem to be very helpful: I would like to show that $f,g$ are chain-homotopic themselves, but if $B$ is acyclic, for example, $q_B$ could be the zero map and so not many informations will be grasped.
