TITLE: What properties do "large topoi" share with actual topoi?
QUESTION [14 upvotes]: Fix Grothendieck universes $\mathcal{U} \in \mathcal{V}$ and suppose that $C$ is a locally $\mathcal{U}$-small category which is $\mathcal{V}$-small. Denote by $Set$ the category of $\mathcal{U}$-small sets (which is also $\mathcal{V}$-small). Suppose that $\mathcal{E}$ is a category which can obtained as a left exact localization of $$Set^{C^{op}}.$$ $\mathcal{E}$ is not (usually) a Grothendieck topos in either universe, but in many ways, it behaves as if it is. It can fail to be locally presentable however.
My question: Exactly which characteristic properties of topoi (e.g. Giraud's axioms, being Cartesian closed...) hold for $\mathcal{E}$ and which do not?

REPLY [12 votes]: If one is not comfortable with Grothendieck universes, then a good heuristic is to replace $\mathcal{U}$-small set by "finite set", and $\mathcal{V}$-small set by "set" (or even in some cases by "countable set"), try to answer analogical questions in this setting, and then transfer answers back to the original setting (this is possible because $\aleph_0$ behaves almost like a strongly inaccessible cardinal).
Here are some trivial observations for finite sets:

$\mathbf{FinSet}^{\aleph_0}$ does not have finite generating family,
$\mathbf{FinSet}^{N}$, where $N$ is the free monoid $\langle 0, +1\rangle$ on a single generator $0$, is not cartesian closed, neither it has a subobject classifier
Every category of the form $\mathbf{FinSet}^{\mathbb{C}^{op}}$ has finite limits and colimits (constructed pointwise); therefore every left exact localization of $\mathbf{FinSet}^{\mathbb{C}^{op}}$ has finite colimits (the same is true for finite limits, but for limits this is not actually a useful heuristic).