Beck-Chevalley for presheaves on groupoids?

[John Baez <baez@math.ucr.edu>]

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