What situations are people aware of where coherence laws can be forced to hold? Say that I have a bunch of categories and some functors between them that compose in some appropriate way only up to natural equivalence; typically, then, one wants to chose the natural equivalences in such away that they satisfy some further coherence relations. For a while, I'd been assuming that there weren't any general situations where you could always do that, but that's not true. The only nontrivial example I know of is the following: given functors F: C -> D and G: D -> C such that FG and GF are both equivalent to the identity functor, one can always choose the relevant natural equivalences in such a way that F and G become adjoint functors (with those natural equivalences as unit and counits). Are there other such situations? Also, what about counterexamples? I'd like to see collections of functors that compose well up to equivalence but where the equivalences can't be chosen in such a way that they satisfy the appropriate coherence laws. (In the above, some people may be bothered by the fact that I'm using words like "appropriate" without making precise what I mean. Here's an example of how to make it precise: say I have a functor from some index category I to the category Decat(1Cat) whose objects are categories and whose morphisms are equivalence classes of functors. When can I lift that to a weak functor from I (considered as a 2-category with only trivial 2-morphisms) to 1Cat (considered as a weak 2-category)? I guess there's always a map from Decat(Hom_{2Cat}(I, 1Cat)) to Hom_{1Cat}(I, Decat(1Cat)); what are its properties, and how do they depend on I? I'm actually interested in somewhat more general situations than that (replacing I by an operad), but the answers where I is a category should give me enough food for thought for a while.) David Carlton carlton@math.stanford.edu