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2 added 36 characters in body

The answer is yes, at least for $2$-fiber products. And fortunately there is an excellent reference online : this is Lemma 52.8.4 in De Jong's stack project. I quote :

"Lemma 8.4. Let C be a site. Let f : X → Y and g : Z → Y be morphisms of ﬁbred categories over C. In this case the stackiﬁcation of the 2-ﬁbre product is the 2-ﬁbre product of the stackiﬁcations."

To get there, and get the proof :

http://stacks.math.columbia.edu/search?keywords=stackification

then select the first answer Lemma 52.8.4 and in the .dvi look for Lemma 8.4. Enjoy !

ps : the permanent tag is Tag 04Y1

1

The answer is yes, at least for $2$-fiber products. And fortunately there is an excellent reference online : this is Lemma 52.8.4 in De Jong's stack project. I quote :

"Lemma 8.4. Let C be a site. Let f : X → Y and g : Z → Y be morphisms of ﬁbred categories over C. In this case the stackiﬁcation of the 2-ﬁbre product is the 2-ﬁbre product of the stackiﬁcations."

To get there, and get the proof :

http://stacks.math.columbia.edu/search?keywords=stackification

then select the first answer Lemma 52.8.4 and in the .dvi look for Lemma 8.4. Enjoy !