Let $C$ be an abelian category and let $X$ be an object with finite length.
Thus there is a composition series
$0=X_0 \stackrel{\iota_0}{\rightarrow}X_1\stackrel{\iota_1}{\rightarrow}\cdots X_{n-1} \stackrel{\iota_{n-1}}{\rightarrow}X_n=X$, where $X_{i}/X_{i-1}$ is simple.
Suppose that we have projective covers $p_i:P_{i}\to X_{i}/X_{i-1}$ for $i=1, \dots, n$. Let $P=\oplus_{i=1}^{n} P_i$.
I want to show that there is an epimorphism from $P$ to $X$. Here is what I have thought: Consider a short exact sequence $0\to X_{i-1} \stackrel{\iota_i}{\rightarrow} X_i \stackrel{j_i}{\rightarrow} X_{i}/X_{i-1} \to 0$. Since we have a projective cover $p_i:P_{i}\to X_{i}/X_{i-1}$, we have a morphism $f_i:P_i \to X_i$ such that $j_i \circ f_i=p_i$.
Define $F_i=\iota_{n-1}\circ \iota_{n-2}\circ \cdots \circ \iota_{i}\circ f_i$ to be a morphism from $P_i$ to $X$. Then define $F:P\to X$ to be $\oplus_{i=1}^n F_i$.
My guess is that this $F$ is epimorphism but I cannot prove it. I appreciate any help.