Hi - One of you must know the answer to this! Suppose we have a weak pullback (= pseudo-pullback) square of groupoids: G -> H | | v v K -> L Suppose we take presheaves on all four. We can get a square hom(G^{op},Set) -> hom(H^{op},Set) ^ ^ | | hom(K^{op},Set) -> hom(L^{op},Set) where the arrows pointing forward - in the same direction as the original arrows - are defined using pushforward, and the arrows pointing backward are defined using pullback. Does this square commute up to natural isomorphism? Do you know a reference somewhere? Some side remarks: 1) This seems related to the "Beck-Chevalley condition". 2) It may work for categories as well as groupoids, but I happen to need it only for groupoids. 3) I really need it with the category Vect replacing Set, so if you know a general result for any sufficiently nice category playing the role of Set here, that would be wonderful. Best, jb