Compatibility of functors with limits

[<leinster@ihes.fr>]

I have the sensation that I'm about to ask a question to which half the
readers of this list will be able to see an answer immediately.
Unfortunately, I'm one of the other half.

What should it mean for a functor to "respect limits"?  Consider the
following informal definition: a functor respects limits if given any
diagram in the domain category, the limit of the image of the diagram is no
bigger than it needs to be.  Formally, let F: A ---> B be a functor, where
B is a category with (for sake of argument) all small limits and colimits.
Let I be a small category and D: I ---> A a diagram in A; write Cone(D) for
the category of cones on D in A, write Cone(FD) for the category of cones
on FD in B, and write

   F_*: Cone(D) ---> Cone(FD)

for the induced functor.  Then F can be said to "respect limits for D" if
the colimit of F_* is the terminal object of Cone(FD) (that is, the limit
cone on FD).

* Example: if D has a limit in A then the limit is a terminal object of
  Cone(D), so F respects limits for D if and only if it preserves the limit
  in the usual sense.

* Example: let B = Set and let A be the category consisting of a pair of
  parallel arrows; a functor F: A ---> B consists of sets and functions

      sigma, tau: F_0 ---> F_1.

  The condition that F respects pullbacks says that sigma and tau are monic
  and that the images of sigma and tau are disjoint.

The thought behind "no bigger than it needs to be" (a very approximate
description, I know) is that if we have a cone on D with vertex v then
there's an induced map from F(v) to lim(FD), which in some sense places a
"lower bound" on lim(FD): e.g. if B = Set and F(v) is nonempty then lim(FD)
is nonempty.  For F to respect limits for D means that lim(FD) is built up
freely from these F(v)s.

So the question is: is this notion of "respecting limits" well-known or
well-understood?  Is there, for instance, some way of rephrasing it that
brings it into more familiar territory?

Thanks very much,
Tom