On the otherhand, since we only need to know that the low dimensional cohomology groups (1 and 2) of $B\Gamma$ vanish, we can probably avoid transfers by looking at explicit interpretations of group cohomology. $ H^1 (\Gamma, Z/2) = Hom(\Gamma, Z/2)$ vanishes if $\Gamma$ is of odd order, so we just need to calculate that there are no non-trivial Z/2 central extensions of $\Gamma$ to show that $H^2$ vanishes. This is probably a standard calculation.

@Oscar: You still need to know that Z/2 cohomology of $B \Gamma$ vanishes, and this is usually proven with transfers. @Jose: Most standard references have a chapter on transfers. The standard reference for me is Allen Hatcher Algebraic Topology which is available online from his website and has a chapter on transfers. (I think he does it for integral cohomology, but the Z/2 case is exactly the same).

Yes. That's exactly right. (Your expression didn't come out, but I know what you mean). There is a "pentagonator". The relevant diagrams are precisely those used to define a tricategory, so can be seen for example in the paper by Gordan-Powers-Street on that subject. Btw, a similar statement holds for algebras. If you have an $A - A\otimes A$ bimodule (and counit which satisfies the pentagon and triangle axioms) then you get an induced monoidal structure on A-Mod. If you add an antipode then these go under the name "Hopfish" algebras. I think this concept is due to Alan Weinstein.

Okay, that's nice. What if you know some additional connectivity for the maps $f_i: X_i \to Y_i$? Would that allow you to conclude more connectivity for the map of geometric realizations? (also, yes, by "simplicial space" I meant simplical object in (nice) topological spaces).

Reid, now you make me feel guilty. Do you not believe in lax monoidal functors? What about the Dold-Kan isomorphism?

As far as the 1-morphisms in Fun(A,B) are concerned, I would be happy to know what happens in either case, i.e. where we use only weak/pseudo transformations or when we use the lax ones.