Mamuka Jibladze wrote:

It just occurred to me that there is something closely related in lattice theory; unfortunately I cannot give a reference, but I remember that one calls a subposet P' of a poset P relatively (co)complete if whenever a subset of P' has an upper bound in P, it has a least upper bound in P'.

This is quite similar, but not the same. I'll take the dual concept (glbs rather than lubs), since it was respect of limits that I wrote about originally. Take an inclusion P' into P of posets, and take a diagram D in P', which might as well be just a subset of P'. Then to say that the inclusion of P' into P respects meets for D is to say that join {lower bounds of D in P'} = meet D where both join and meet are taken in P. (I'm assuming that P is complete; if not, respect of meets for D also asserts that the join and the meet exist.) In my first mail I described, vaguely, respect of limits as meaning that the limit of the image is "no bigger than it needs to be". Order theory is (unsurprisingly) the context in which this makes the most sense: the greatest lower bound of D in P obviously needs to be greater than all the lower bounds of D in P', but that understood, it's minimal. The dual of Mamuka's statement is that, with D and P' and P as above, if D has a lower bound in P then it has a greatest lower bound in P'. Here's an example where meets are respected but this condition (= relative completeness?) fails. Let 0 be the empty category. For any category C, the unique functor 0 ---> C respects limits if and only if C has an object that is both initial and terminal. So if 1 is the one-element lattice then 0 ---> 1 respects meets, and the subset 0 of 0 has a lower bound in 1 but no lower bound in 0. Tom