Lemma 12.5.10. Let $\mathcal{A}$ be an abelian category. Let $0 \to A \to B \to C \to 0$ be a short exact sequence.

Given a morphism $s : C \to B$ left inverse to $B \to C$, there exists a unique $\pi : B \to A$ such that $(s, \pi )$ splits the short exact sequence as in Definition 12.5.9.

Given a morphism $\pi : B \to A$ right inverse to $A \to B$, there exists a unique $s : C \to B$ such that $(s, \pi )$ splits the short exact sequence as in Definition 12.5.9.

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