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 a Grothendieck topos in either universe, but in many ways, it behaves as if it is. It is not 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?

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?