By definition, a $\mathbf{U}$-pretopos is a category $\mathcal{C}$ that satisfies Giraud's axioms except for the existence of topological generators, i.e.
- $\mathcal{C}$ has finite limits,
- $\mathcal{C}$ is a $\mathbf{U}$-extensive category, i.e. $\mathcal{C}$ has coproducts for $\mathbf{U}$-small families of objects, and coproducts are disjoint and stable under pullback, and
- $\mathcal{C}$ is an effective regular (= Barr exact) category, i.e. $\mathcal{C}$ has quotients for equivalence relations, and these are effective and stable under pullback.
Let $\mathbf{Set}$ be the category of $\mathbf{U}$-sets. It is clear that $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is always a $\mathbf{U}$-pretopos, regardless of whether $\mathcal{A}$ is $\mathbf{U}$-small or not. And because the definition of $\mathbf{U}$-pretopos involves only colimits and finite limits, any "left exact localisation" (= reflective subcategory with finite-limit-preserving reflector) of a $\mathbf{U}$-pretopos is again a $\mathbf{U}$-pretopos.
By Giraud's theorem, the following are equivalent for a $\mathbf{U}$-pretopos $\mathcal{E}$:
- $\mathcal{E}$ is a Grothendieck $\mathbf{U}$-topos.
- $\mathcal{E}$ is a locally presentable $\mathbf{U}$-category.
- $\mathcal{E}$ has a $\mathbf{U}$-small separating family.
Thus any $\mathbf{U}$-pretopos that fails to be a Grothendieck $\mathbf{U}$-topos is necessarily not locally presentable. This can happen even if it is locally $\mathbf{U}$-small: for instance, take $\mathcal{A}$ to be a one-object groupoid that is not $\mathbf{U}$-small and consider $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$. (In fact, this is even an elementary topos!)
Now, if $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is locally $\mathbf{U}$-small, then it is cartesian closed (using the essentially same construction as the case when $\mathcal{A}$ is $\mathbf{U}$-small). And if $\mathcal{E}$ is a reflective subcategory of $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ with a reflector that preserves finite products, then $\mathcal{E}$ is cartesian closed and inherits exponential objects from $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$. (In fact, $\mathcal{E}$ is an exponential ideal in this case.)
On the other hand, if $\mathcal{A}$ has the property that each slice category $\mathcal{A}_{/ a}$ is essentially $\mathbf{U}$-small (e.g. $\mathcal{A} = \mathbf{Ord}$), then $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ has a subobject classifier (using essentially the same construction as the case when $\mathcal{A}$ is $\mathbf{U}$-small). This is not a necessary condition: cf. the example where $\mathcal{A}$ is a "big group".
Finally, let me remind you of an important theorem in the theory of elementary toposes: if $\mathcal{F}$ is an elementary topos and $\mathcal{E}$ is a reflective subcategory of $\mathcal{F}$ with finite-limit-preserving reflector, then $\mathcal{E}$ is also an elementary topos.
