I'm trying to prove that given a morphism of chain complexes $p : X \rightarrow Y$ where all the $p_n : X_n \rightarrow Y_n$ are surjective, one has a lift
$$\require{AMScd} \begin{CD} 0 @>>> X\\ @VVV @V{p}VV \\ C @>>> Y \end{CD}$$
where $C$ is acyclic ( $C_n$ is projective for all $n$ and $C_n = 0$ for $n \ll 0$ )
Any solution would be very much appreciated.