Let $C$ be a site and $CAT$ the 2-category of categories. Given a contravariant 2-functor $A:C\rightarrow CAT$, we can of course consider the associated stack. This is done by first considering the associated prestack, denoted by $\hat{A}$, (i.e. sheafifying the hom) and then considering $2$-$lim_\mathfrak{U}Des(\mathfrak{U},\hat{A})$, the descent data of $\hat{A}$ with respect to the covering $\mathfrak{U}$.

EDIT: My question is, could we stackify $A$ purely with the descent data? In other words, define $A'(U):=2$-$lim_\mathfrak{U}(Des(\mathfrak{U},A))$,

$A''(U):=2$-$lim_\mathfrak{U}(Des(\mathfrak{U},A'))$ and

$A'''(U):=2$-$lim_\mathfrak{U}(Des(\mathfrak{U},A''))$.

Would $A'''$ be equivalent to the associated stack of $A$?.

Thank you in advance.

Higher topos theory, §6.5.3]. $\endgroup$ – Zhen Lin Oct 11 '14 at 12:52