The exercise is Let $\mathcal{A}$ be a small abelian category. If $[A_1] = [A_2]$ in $K_0(\mathcal{A})$, show that there are short exact sequences in $\mathcal{A}$ $0 \rightarrow C'\rightarrow C_1 \rightarrow C'' \rightarrow 0, \;\;0 \rightarrow C'\rightarrow C_2 \rightarrow C'' \rightarrow 0$ such that $A_1 \oplus C_1 \cong A_2 \oplus C_2.$
Hint: First find sequences $0 \rightarrow D_i'\rightarrow D_i \rightarrow D_i'' \rightarrow 0$ such that $A_1 \oplus D_1' \oplus D_1'' \oplus D_2 \cong A_2 \oplus D_2' \oplus D_2'' \oplus D_1,$ and set $C_i=D_i'\oplus D_i'' \oplus D_j.$
I have tried to construct the exact sequence given in the hint with component wise direct sums of $A_1$ and $A_2$ but even if I am able to satisfy the isomorphism the exact sequences fall apart and vice versa. So I am stuck, if you could please give me some hint I would try my best. I give a small description of my attempt below,
Since my end goal is to achieve the isomorphism $A_1 \oplus D_1' \oplus D_1'' \oplus D_2 \cong A_2 \oplus D_2' \oplus D_2'' \oplus D_1,$ first I fix $D_1' = A_2$ and $D_2' = A_1$, now I was trying to make adjustments later but as mentioned above I was failing to construct the exact sequence. If could please help me out it would e great. Thank you.