Exponential 2 categorical

If \(S^T\) exists, it is automatically a \(2\)-categorical exponential. Indeed, for every topos \(U\), the arrow topos \(\Ar(U)\) represents arrows between geometric morphisms out of \(U\), and \begin{align*} \Hom_{\Cat}([1],\Geom(U,S^T)) &\iso \Hom_{\Topos}(\Ar(U),S^T) \\ &\iso \Hom_{\Topos}(\Ar(U) \otimes T, S) \\ &\iso \Hom_{\Topos}(\Ar(U \otimes T),S) \\ &\iso \Hom_{\Cat}([1],\Geom(U \otimes T,S)). \end{align*} Together with the original universal property on objects, this identifies the entire Hom categories. Equivalently, one may repeat the same argument with the topos of \([n]\)-diagrams for every \(n\) and use the complete Segal description of a category.