The Stacks project

Situation 7.18.1. Here we are given

1. a cofiltered index category $\mathcal{I}$,

2. for $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ a site $\mathcal{C}_ i$ such that every covering in $\mathcal{C}_ i$ has a finite index set,

3. for a morphism $a : i \to j$ in $\mathcal{I}$ a morphism of sites $f_ a : \mathcal{C}_ i \to \mathcal{C}_ j$ given by a continuous functor $u_ a : \mathcal{C}_ j \to \mathcal{C}_ i$,

such that $f_ a \circ f_ b = f_ c$ whenever $c = a \circ b$ in $\mathcal{I}$.