This kind of descent is described in the article stack using the bar construction to write a sieve as a colimit of representables.

Let $C$ be a site, $p:D\to C$ a fibration, $x\in {C}_{0}$, ${U}_{x}$ a cover of $x$. ${h}_{x}\hookrightarrow C/x$ the full subcategory of arrows, that factor through some $u\to x$ of the cover; the corresponding functor ${C}^{\mathrm{op}}\to Cat$ is then a sieve. The category of descent data (sometimes shortcut to descent category) of $p$ along ${U}_{x}$ is defined to be the functor category $[{h}_{x},D]$.