Lemma 57.10.4. Let $\mathcal{A}$ be an abelian category with enough negative objects. Let $X \in D^ b(\mathcal{A})$. Let $b \in \mathbf{Z}$ with $H^ i(X) = 0$ for $i > b$. Then there exists a map $N[-b] \to X$ such that the induced map $N \to H^ b(X)$ is surjective and $\mathop{\mathrm{Hom}}\nolimits (H^ b(X), N) = 0$.

**Proof.**
Using the truncation functors we can represent $X$ by a complex $A^ a \to A^{a + 1} \to \ldots \to A^ b$ of objects of $\mathcal{A}$. Choose $N$ in $\mathcal{A}$ such that there exists a surjection $t : N \to A^ b$ and such that $\mathop{\mathrm{Hom}}\nolimits (A^ b, N) = 0$. Then the surjection $t$ defines a map $N[-b] \to X$ as desired.
$\square$

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