Following Fred's lead, I discovered the following in Lawvere's thesis: (reworded to make the language more modern). Suppose u: C --> D is a functor such that C is small (I'm not sure it matters), every coterminous pair of maps in D is the front half of a commutative square and u is cofinal in the sense that for any d in D, there is a c in C and a map uc --> d. Then for any functor f: D --> E any limit of fu is a limit of f. These conditions are readily seen to be satisfied for the inclusion of the diagonal into the double sequence, so Fred's memory is completely accurate on this. Michael =================================