If G -> H | <= | v v K -> L is a square (with a 2-cell as shown) in cat then applying ^ =((-)^{op},Set) gives G^<- H^ ^ => ^ | | K^<- L^ and taking the mate with respect to the horizontal adjunctions given by left Kan extension gives G^-> H^ ^ => ^ | | K^-> L^ If the original square is a comma square then the 2-cell in the third square is invertible. There are squares other than comma squares, cocomma squares for example, for which the 2-cell in the third square is invertible. See Rene Guitart's early work on exact squares. Best, Rj Wood

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