Lemma 48.30.2. The result of Lemma 48.30.1 holds even for $L \in D^+_{\textit{Coh}}(\mathcal{O}_ X)$.

**Proof.**
Namely, if $(K_ n)$ is a Deligne system then there exists a $b \in \mathbf{Z}$ such that $H^ i(K_ n) = 0$ for $i > b$. Then $\mathop{\mathrm{Hom}}\nolimits (K_ n, L) = \mathop{\mathrm{Hom}}\nolimits (K_ n, \tau _{\leq b}L)$ and $\mathop{\mathrm{Hom}}\nolimits (K, L) = \mathop{\mathrm{Hom}}\nolimits (K, \tau _{\leq b}L)$. Hence using the result of the lemma for $\tau _{\leq b}L$ we win.
$\square$

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