@S.carmeli I am a little confused by what you mean. $X_{/U} \to X$ is a geometric morphism, and in this particular case all limits are preserved by the inverse image half of the geometric morphism. Or am I missing something here?

@AdrianClough I think he means that $X\to X_{/U}$ is not a geometric morphism (whose right adjoint would be $f^*$, and left adjoint $f_!$).

@CharlesRezk Oh, ok!

@DenisNardin Thanks, this is an interesting point. Finite meets in locales can viewed as finite products, finite pullbacks, or arbitrary finite limits, so it's funny how, if you want to generalise from locales to toposes, you don't know a priori which sort of finite limits your inverse image functors are supposed to preserve. Only a posteriori do you discover that your finite meets where finite limits.

@AdrianClough Well, one way of thinking about this is that you want to preserve truncatedness (because you want your map of topos to restrict to a map of underlying locales), and left exact functors are the natural object that preserve truncatedness

(in fact preserving the terminal object can be thought of "preserving (-2)-truncated objects", although I'm unsure if this is actually convincing)

@DenisNardin Ah! This is a very nice observation! This is precisely the sort of answer I was hoping for: Every ($\infty$-)topos has property "bla" that is not preserved if (in addition to pullbacks) we do not preserve final objects.

Well, it's really saying the same thing (as I said, terminal objects are (-2)-truncated objects), but it might be a more "geometric" way of thinking about it

Right. I think this is very nice, because truncatedness plays such a central role, e.g. for Postnikov towers; this is structure that I definitely want to preserve.

@YuriSulyma great, thanks -- i should have guessed

@CharlesRezk indeed this was what I meant thanks.