Lemma 12.5.8. Let $\mathcal{A}$ be an abelian category. Let $0 \to M_1 \to M_2 \to M_3 \to 0$ be a complex of $\mathcal{A}$.

$M_1 \to M_2 \to M_3 \to 0$ is exact if and only if

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(M_3, N) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(M_2, N) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(M_1, N) \]is an exact sequence of abelian groups for all objects $N$ of $\mathcal{A}$, and

$0 \to M_1 \to M_2 \to M_3$ is exact if and only if

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(N, M_1) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(N, M_2) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(N, M_1) \]is an exact sequence of abelian groups for all objects $N$ of $\mathcal{A}$.

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