I am reading Lurie's Tannakian paper (<http://www.math.harvard.edu/~lurie/papers/Tannaka.pdf>) and I am confused about one point.
At the end of page 3 he defines a stack-hom in any topos, which is defined by stackification. Now, I know that there are problems with sheafification in the fpqc topology. Assume the topos is fpqc sheaves on Spec Z, shouldn't there be a problem with stackifying things?


Has this stackification nothing to do with fpqc sheafification?


Is it that this particular stackificaion does not cause any problems?


I would be glad if someone could clear this all up for me.



fpqc stackification
Summing up the comments: stackification works in any topos; fpqc sheaves do not form a topos (size does matter).


