Situation 20.41.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. We are given

a collection of opens $\mathcal{B}$ of $X$,

for $U \in \mathcal{B}$ an object $K_ U$ in $D(\mathcal{O}_ U)$,

for $V \subset U$ with $V, U \in \mathcal{B}$ an isomorphism $\rho ^ U_ V : K_ U|_ V \to K_ V$ in $D(\mathcal{O}_ V)$,

such that whenever we have $W \subset V \subset U$ with $U, V, W$ in $\mathcal{B}$, then $\rho ^ U_ W = \rho ^ V_ W \circ \rho ^ U_ V|_ W$.

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