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.



Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first?
The three-step process is given as Theorem 3.8 in Ross Street, *Two dimensional sheaf theory*, J. Pure Appl. Algebra **23** (1982) 251–270 under some smallness assumptions. Without some smallness hypotheses, you may run into set-theoretic problems. For example, Waterhouse produced a presheaf with no flat sheafification for Theorem 5.5 in *Basically bounded functors and flat sheaves* Pac. J. Math. **57** no. 2 (1975) 597-610.


There is a one-step process using hypercovers instead of covers. As Zhen Lin mentioned in a comment, this is discussed in the beginning of section 6.5.3 in Lurie's *Higher Topos Theory*.


