Excercise 2.2.1 in Weibel ("An Introduction to Homological Algebra") states that an object $P$ in the category of chain complexes over an abelian category is projective if and only it is a split exact complex of projectives.

I was able to solve the only-if-part but I have touble with the if-part and would be glad if someone can give me some help. This is no homework!

**What have I a tried so far ?** Given an epimorphism $\pi: X \to Y$ and a morphism $f: P \to Y$, it has to be shown that there is a morphism $g: P \to X$ s.t. $\pi \circ g=f$.

Weibel hints to consider the special case $0 \to P_1 \cong P_0 \to 0$. It's easy to construct $g$ in this case: $\pi$ epi means that each $\pi_i:X_i \to Y_i$ is epi. By projectivity of $P_1$ there is a hom. $g_1: P_1 \to X_1$ s.t. $\pi_1 \circ g_1 = f_1$. If $d^P$ resp. $d^X$ denotes the differential in $P$ resp. $X$, set $$g_0 := d^X_1 \circ g_1 \circ (d^P_1)^{-1}: P_0 \to X_0,\qquad g_i = 0: P_i \to X_i\; (i\neq 0,1)$$ Then $g=(g_i): P \to X$ is a morphisms s.t. $\pi \circ g=f$.

But I have no idea how to generalize this procedure to the general case where $d^P$ can not be expected to be an isomorphism.