Lemma 48.30.3. Let $j : U \to X$ be an open immersion of Noetherian schemes.

Let $(K_ n)$ and $(L_ n)$ be Deligne systems. Let $K$ and $L$ be the values of the constant systems $(K_ n|_ U)$ and $(L_ n|_ U)$. Given a morphism $\alpha : K \to L$ of $D(\mathcal{O}_ U)$ there is a unique morphism of pro-systems $(K_ n) \to (L_ n)$ of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ whose restriction to $U$ is $\alpha $.

Given $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ U)$ there exists a Deligne system $(K_ n)$ such that $(K_ n|_ U)$ is constant with value $K$.

The pro-object $(K_ n)$ of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ of (2) is unique up to unique isomorphism (as a pro-object).

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