The Stacks project

Situation 85.24.1. Let $(\mathcal{C}, \mathcal{O}_\mathcal {C})$ be a ringed site. We are given

1. a category $\mathcal{B}$ and a functor $u : \mathcal{B} \to \mathcal{C}$,

2. an object $E_ U$ in $D(\mathcal{O}_{u(U)})$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$,

3. an isomorphism $\rho _ a : E_ U|_{\mathcal{C}/u(V)} \to E_ V$ in $D(\mathcal{O}_{u(V)})$ for $a : V \to U$ in $\mathcal{B}$

such that whenever we have composable arrows $b : W \to V$ and $a : V \to U$ of $\mathcal{B}$, then $\rho _{a \circ b} = \rho _ b \circ \rho _ a|_{\mathcal{C}/u(W)}$.