Just a quick comment on Steve Vickers' post. This ought to be well known, but either it isn't as well known as I thought, or people aren't thinking of it in this connection. If C --> D induces an equivalence between idempotent completions (for example, if it is inclusion of C into its idempotent completion), then the induced Set^D --> Set^C is an equivalence. In fact, the analagous statement is true for any base that is itself idempotent complete. Now of course an inclusion of monoids that had that property would already be an equivalence since the effect of idempotent completion is to add more objects, but (effectively) no more arrows. On the other hand, for a monoid with many objects (aka a category), the situation is quite different. On the other hand, it might be useful to confine the discussion to idempotent complete categories to avoid this particular problem. Michael ==============================================================================