Definition 2.1.
Let \(I\) be a small category, and let \(T\) be a category with pullbacks and \(I\)-indexed colimits. We say that \(T\) satisfies descent for \(I\)-indexed colimits, or that \(I\)-indexed colimits are van Kampen, if the functor \begin{align*} T\catop \to \Cat, \qquad X \mapsto T_{/X} \end{align*} takes \(I\)-indexed colimit diagrams in \(T\) to limit diagrams in \(\Cat\). Explicitly, for every diagram \(X_{\bullet}\colon I \to T\) the canonical functor
\[T_{/\,\colim_{i \in I}X_i} \longrightarrow \lim_{i \in I\catop} T_{/X_i}\]
is an equivalence.