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If $E$ is a topos, is there a nice way to characterize the category of logical morphisms $E\to Set$? Is it complete and/or cocomplete?

The topos $Set$ geometrically represents a point; what does it logically represent?

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    $\begingroup$ I'd be interested if there were any good answers to this. A related question is mathoverflow.net/questions/4044/logical-endofunctors-of-set where even in the case $E = Set$, existence of non-trivial logical endofunctors involve fairly large cardinal hypotheses. $\endgroup$
    – Todd Trimble
    Commented Sep 12, 2010 at 1:20
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    $\begingroup$ What is a logical morphism? $\endgroup$
    – Martin Brandenburg
    Commented Sep 13, 2010 at 9:54
  • $\begingroup$ @MartinBrandenburg: Roughly, a functor $F:A\to B$ between toposes is called "logical" if it preserves finite limits and power-objects. See ncatlab.org/nlab/show/logical+functor $\endgroup$
    – David Spivak
    Commented May 8, 2014 at 18:22

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