TITLE: Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first?
QUESTION [6 upvotes]: 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.

REPLY [9 votes]: 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.