Suppose given a diagram X > Z < Y of Gspaces (G a discrete group). Is (X ×^{h}_{Z} Y)_{hG} weakly equivalent to X_{hG} ×^{h}_{ZhG} Y_{hG}?
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Yes. A sketch: Taking products with the free Gspace EG commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibration. Having done this, there is a natural long exact sequence of homotopy groups > pi_* (U) > pi_*(U_{hG}) > pi_*(BG) > ... and applying this to the pullback diagram you can deduce (from the 5lemma) that the natural map from the orbit of pullbacks to the pullback of the orbits is a weak equivalence. 

