Lemma 12.22.4. Let $\mathcal{A}$ be an abelian category. Let $0 \to (A, d) \to (B, d) \to (C, d) \to 0$ be a short exact sequence of differential objects. Then we get an exact homology sequence

\[ \ldots \to H(C, d) \to H(A, d) \to H(B, d) \to H(C, d) \to \ldots \]

**Proof.**
Apply Lemma 12.13.12 to the short exact sequence of complexes

\[ \begin{matrix} 0
& \to
& A
& \to
& B
& \to
& C
& \to
& 0
\\ & & \downarrow
& & \downarrow
& & \downarrow
\\ 0
& \to
& A
& \to
& B
& \to
& C
& \to
& 0
\\ & & \downarrow
& & \downarrow
& & \downarrow
\\ 0
& \to
& A
& \to
& B
& \to
& C
& \to
& 0
\end{matrix} \]

where the vertical arrows are $d$. $\square$

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