Definition 59.51.1. Let $I$ be a preordered set. Let $(X_ i, f_{i'i})$ be an inverse system of schemes over $I$. A *system $(\mathcal{F}_ i, \varphi _{i'i})$ of sheaves on $(X_ i, f_{i'i})$* is given by

a sheaf $\mathcal{F}_ i$ on $(X_ i)_{\acute{e}tale}$ for all $i \in I$,

for $i' \geq i$ a map $\varphi _{i'i} : f_{i'i}^{-1}\mathcal{F}_ i \to \mathcal{F}_{i'}$ of sheaves on $(X_{i'})_{\acute{e}tale}$

such that $\varphi _{i''i} = \varphi _{i''i'} \circ f_{i'' i'}^{-1}\varphi _{i'i}$ whenever $i'' \geq i' \geq i$.

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