In the proof of lemma 3.2 in this PDF, it said:
Let $0 → A → A′ → A′′ → 0$ be a short exact sequence. $P′′_{*}→ A′′$ be a projective resolution. (i.e. $ ··· P_{1}′′→ P_{0}′′ → A′′ → 0$ is exact with $P_{i}$ projective)
Now let $α′\in A′$. If $α′ \notin Ker(A′ → A′′)$ then it has nonzero image $α′′ ∈ A′′$, and there is some $p′′ ∈ P_{0}′′$ mapping onto $α′′$. By commutativity, this $p′′$ maps onto $α′$.
Why $p′′$ must maps onto $α′$ but not other elements which also has $α′′$ as its image?